q-Tensor Squares of Polycyclic Groups Nora R. Rocco Universidade - PowerPoint PPT Presentation

ν q ( G ) and some relevant sections ν q ( G ) := ( ν ( G ) ∗ F ( � G )) / J For q = 0 the set of all relations (1) to (6) is empty; in this case ν ( G ) ∗ F ( � G ) / J ∼ = ν ( G ) G and G ϕ are embedded into ν q ( G ) , for all q ≥ 0 Set T = [ G , G ϕ ] ≤ ν q ( G ) and let G be the subgroup of ν q ( G ) generated by ( the images of) � G The subgroup Υ q ( G ) = T G is normal in ν q ( G ) and ν q ( G ) = G ϕ · ( G · Υ q ( G )) N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

ν q ( G ) and some relevant sections ν q ( G ) := ( ν ( G ) ∗ F ( � G )) / J For q = 0 the set of all relations (1) to (6) is empty; in this case ν ( G ) ∗ F ( � G ) / J ∼ = ν ( G ) G and G ϕ are embedded into ν q ( G ) , for all q ≥ 0 Set T = [ G , G ϕ ] ≤ ν q ( G ) and let G be the subgroup of ν q ( G ) generated by ( the images of) � G The subgroup Υ q ( G ) = T G is normal in ν q ( G ) and ν q ( G ) = G ϕ · ( G · Υ q ( G )) N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

ν q ( G ) and some relevant sections ν q ( G ) := ( ν ( G ) ∗ F ( � G )) / J For q = 0 the set of all relations (1) to (6) is empty; in this case ν ( G ) ∗ F ( � G ) / J ∼ = ν ( G ) G and G ϕ are embedded into ν q ( G ) , for all q ≥ 0 Set T = [ G , G ϕ ] ≤ ν q ( G ) and let G be the subgroup of ν q ( G ) generated by ( the images of) � G The subgroup Υ q ( G ) = T G is normal in ν q ( G ) and ν q ( G ) = G ϕ · ( G · Υ q ( G )) N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

ν q ( G ) and G ⊗ q G (Ellis [1995], Bueno [2006]) Υ q ( G ) ∼ = G ⊗ q G , for all q ≥ 0 . The q - exterior square G ∧ q G , is the quotient of G ⊗ q G by its subgroup � k ⊗ k | k ∈ G � . Thus, G ∧ q G ∼ = Υ q ( G ) / ∆ q ( G ) , where ∆ q ( G ) = � [ g , g ϕ ] | g ∈ G � . Let ρ ′ : Υ q ( G ) → G be induced by [ g , h ϕ ] �→ [ g , h ] , � k �→ k q , for all g , h , k ∈ G . ⇒ Ker ( ρ ′ ) / ∆ q ( G ) ∼ = H 2 ( G , Z q ) . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

ν q ( G ) and G ⊗ q G (Ellis [1995], Bueno [2006]) Υ q ( G ) ∼ = G ⊗ q G , for all q ≥ 0 . The q - exterior square G ∧ q G , is the quotient of G ⊗ q G by its subgroup � k ⊗ k | k ∈ G � . Thus, G ∧ q G ∼ = Υ q ( G ) / ∆ q ( G ) , where ∆ q ( G ) = � [ g , g ϕ ] | g ∈ G � . Let ρ ′ : Υ q ( G ) → G be induced by [ g , h ϕ ] �→ [ g , h ] , � k �→ k q , for all g , h , k ∈ G . ⇒ Ker ( ρ ′ ) / ∆ q ( G ) ∼ = H 2 ( G , Z q ) . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

ν q ( G ) and G ⊗ q G (Ellis [1995], Bueno [2006]) Υ q ( G ) ∼ = G ⊗ q G , for all q ≥ 0 . The q - exterior square G ∧ q G , is the quotient of G ⊗ q G by its subgroup � k ⊗ k | k ∈ G � . Thus, G ∧ q G ∼ = Υ q ( G ) / ∆ q ( G ) , where ∆ q ( G ) = � [ g , g ϕ ] | g ∈ G � . Let ρ ′ : Υ q ( G ) → G be induced by [ g , h ϕ ] �→ [ g , h ] , � k �→ k q , for all g , h , k ∈ G . ⇒ Ker ( ρ ′ ) / ∆ q ( G ) ∼ = H 2 ( G , Z q ) . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

ν q ( G ) and G ⊗ q G (Ellis [1995], Bueno [2006]) Υ q ( G ) ∼ = G ⊗ q G , for all q ≥ 0 . The q - exterior square G ∧ q G , is the quotient of G ⊗ q G by its subgroup � k ⊗ k | k ∈ G � . Thus, G ∧ q G ∼ = Υ q ( G ) / ∆ q ( G ) , where ∆ q ( G ) = � [ g , g ϕ ] | g ∈ G � . Let ρ ′ : Υ q ( G ) → G be induced by [ g , h ϕ ] �→ [ g , h ] , � k �→ k q , for all g , h , k ∈ G . ⇒ Ker ( ρ ′ ) / ∆ q ( G ) ∼ = H 2 ( G , Z q ) . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

ν q ( G ) and G ⊗ q G (Ellis [1995], Bueno [2006]) Υ q ( G ) ∼ = G ⊗ q G , for all q ≥ 0 . The q - exterior square G ∧ q G , is the quotient of G ⊗ q G by its subgroup � k ⊗ k | k ∈ G � . Thus, G ∧ q G ∼ = Υ q ( G ) / ∆ q ( G ) , where ∆ q ( G ) = � [ g , g ϕ ] | g ∈ G � . Let ρ ′ : Υ q ( G ) → G be induced by [ g , h ϕ ] �→ [ g , h ] , � k �→ k q , for all g , h , k ∈ G . ⇒ Ker ( ρ ′ ) / ∆ q ( G ) ∼ = H 2 ( G , Z q ) . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Cyclic Groups T.Bueno & N.R. [2011] Let d = gcd ( q , n ) . Then C ∞ ⊗ q C ∞ ∼ = C ∞ × C q ,   C n × C d , if d is odd ,  C n ⊗ q C n ∼ = C n × C d , if d is even and either 4 | n or 4 | q ;   C 2 n × C d / 2 , otherwise. N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Cyclic Groups T.Bueno & N.R. [2011] Let d = gcd ( q , n ) . Then C ∞ ⊗ q C ∞ ∼ = C ∞ × C q ,   C n × C d , if d is odd ,  C n ⊗ q C n ∼ = C n × C d , if d is even and either 4 | n or 4 | q ;   C 2 n × C d / 2 , otherwise. N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Direct Products T.Bueno & N.R. [2011] Let G = N × H , N = N / N ′ N q and H = H / H ′ H q . Then (i) ν q ( G ) = � N , N ϕ , � N � × [ N , H ϕ ][ H , N ϕ ] × � H , H ϕ , � H� ; (ii) � H , H ϕ , � H� ∼ � N , N ϕ , � N � ∼ = ν q ( H ); = ν q ( N ) . (iii) Υ q ( G ) = Υ q ( N ) × [ N , H ϕ ][ H , N ϕ ] × Υ q ( H ); (iv) [ N , H ϕ ] ∼ = N ⊗ Z q H ∼ = [ H , N ϕ ] . (v) ∆ q ( N × H ) = ∆ q ( N ) × ∆( H ) × U , where U = � [ x , y ϕ ][ y , x ϕ ] | x ∈ N , y ∈ H � . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Direct Products T.Bueno & N.R. [2011] Let G = N × H , N = N / N ′ N q and H = H / H ′ H q . Then (i) ν q ( G ) = � N , N ϕ , � N � × [ N , H ϕ ][ H , N ϕ ] × � H , H ϕ , � H� ; (ii) � H , H ϕ , � H� ∼ � N , N ϕ , � N � ∼ = ν q ( H ); = ν q ( N ) . (iii) Υ q ( G ) = Υ q ( N ) × [ N , H ϕ ][ H , N ϕ ] × Υ q ( H ); (iv) [ N , H ϕ ] ∼ = N ⊗ Z q H ∼ = [ H , N ϕ ] . (v) ∆ q ( N × H ) = ∆ q ( N ) × ∆( H ) × U , where U = � [ x , y ϕ ][ y , x ϕ ] | x ∈ N , y ∈ H � . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Direct Products T.Bueno & N.R. [2011] Let G = N × H , N = N / N ′ N q and H = H / H ′ H q . Then (i) ν q ( G ) = � N , N ϕ , � N � × [ N , H ϕ ][ H , N ϕ ] × � H , H ϕ , � H� ; (ii) � H , H ϕ , � H� ∼ � N , N ϕ , � N � ∼ = ν q ( H ); = ν q ( N ) . (iii) Υ q ( G ) = Υ q ( N ) × [ N , H ϕ ][ H , N ϕ ] × Υ q ( H ); (iv) [ N , H ϕ ] ∼ = N ⊗ Z q H ∼ = [ H , N ϕ ] . (v) ∆ q ( N × H ) = ∆ q ( N ) × ∆( H ) × U , where U = � [ x , y ϕ ][ y , x ϕ ] | x ∈ N , y ∈ H � . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Direct Products T.Bueno & N.R. [2011] Let G = N × H , N = N / N ′ N q and H = H / H ′ H q . Then (i) ν q ( G ) = � N , N ϕ , � N � × [ N , H ϕ ][ H , N ϕ ] × � H , H ϕ , � H� ; (ii) � H , H ϕ , � H� ∼ � N , N ϕ , � N � ∼ = ν q ( H ); = ν q ( N ) . (iii) Υ q ( G ) = Υ q ( N ) × [ N , H ϕ ][ H , N ϕ ] × Υ q ( H ); (iv) [ N , H ϕ ] ∼ = N ⊗ Z q H ∼ = [ H , N ϕ ] . (v) ∆ q ( N × H ) = ∆ q ( N ) × ∆( H ) × U , where U = � [ x , y ϕ ][ y , x ϕ ] | x ∈ N , y ∈ H � . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Direct Products T.Bueno & N.R. [2011] Let G = N × H , N = N / N ′ N q and H = H / H ′ H q . Then (i) ν q ( G ) = � N , N ϕ , � N � × [ N , H ϕ ][ H , N ϕ ] × � H , H ϕ , � H� ; (ii) � H , H ϕ , � H� ∼ � N , N ϕ , � N � ∼ = ν q ( H ); = ν q ( N ) . (iii) Υ q ( G ) = Υ q ( N ) × [ N , H ϕ ][ H , N ϕ ] × Υ q ( H ); (iv) [ N , H ϕ ] ∼ = N ⊗ Z q H ∼ = [ H , N ϕ ] . (v) ∆ q ( N × H ) = ∆ q ( N ) × ∆( H ) × U , where U = � [ x , y ϕ ][ y , x ϕ ] | x ∈ N , y ∈ H � . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Direct Products T.Bueno & N.R. [2011] Let G = N × H , N = N / N ′ N q and H = H / H ′ H q . Then (i) ν q ( G ) = � N , N ϕ , � N � × [ N , H ϕ ][ H , N ϕ ] × � H , H ϕ , � H� ; (ii) � H , H ϕ , � H� ∼ � N , N ϕ , � N � ∼ = ν q ( H ); = ν q ( N ) . (iii) Υ q ( G ) = Υ q ( N ) × [ N , H ϕ ][ H , N ϕ ] × Υ q ( H ); (iv) [ N , H ϕ ] ∼ = N ⊗ Z q H ∼ = [ H , N ϕ ] . (v) ∆ q ( N × H ) = ∆ q ( N ) × ∆( H ) × U , where U = � [ x , y ϕ ][ y , x ϕ ] | x ∈ N , y ∈ H � . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

f.g. Abelian Groups If A = � r i = 1 C i , where C i = � x i � , then r � � [ C j , C ϕ k ][ C k , C ϕ Υ q ( A ) = Υ q ( C i ) × j ] . i = 1 1 ≤ j < k ≤ r Here we have Υ q ( C i ) = � [ x i , x ϕ i ] , � x i � and [ C j , C ϕ k ][ C k , C ϕ j ] = � [ x j , x ϕ k ][ x k , x ϕ j ] , [ x j , x ϕ k ] � . In addition, ∆ q ( A ) = � r i ] � × � i = 1 � [ x i , x ϕ j < k � [ x j , x ϕ k ][ x k , x ϕ j ] � . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

f.g. Abelian Groups If A = � r i = 1 C i , where C i = � x i � , then r � � [ C j , C ϕ k ][ C k , C ϕ Υ q ( A ) = Υ q ( C i ) × j ] . i = 1 1 ≤ j < k ≤ r Here we have Υ q ( C i ) = � [ x i , x ϕ i ] , � x i � and [ C j , C ϕ k ][ C k , C ϕ j ] = � [ x j , x ϕ k ][ x k , x ϕ j ] , [ x j , x ϕ k ] � . In addition, ∆ q ( A ) = � r i ] � × � i = 1 � [ x i , x ϕ j < k � [ x j , x ϕ k ][ x k , x ϕ j ] � . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

f.g. Abelian Groups If A = � r i = 1 C i , where C i = � x i � , then r � � [ C j , C ϕ k ][ C k , C ϕ Υ q ( A ) = Υ q ( C i ) × j ] . i = 1 1 ≤ j < k ≤ r Here we have Υ q ( C i ) = � [ x i , x ϕ i ] , � x i � and [ C j , C ϕ k ][ C k , C ϕ j ] = � [ x j , x ϕ k ][ x k , x ϕ j ] , [ x j , x ϕ k ] � . In addition, ∆ q ( A ) = � r i ] � × � i = 1 � [ x i , x ϕ j < k � [ x j , x ϕ k ][ x k , x ϕ j ] � . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

f.g. Abelian Groups If A = � r i = 1 C i , where C i = � x i � , then r � � [ C j , C ϕ k ][ C k , C ϕ Υ q ( A ) = Υ q ( C i ) × j ] . i = 1 1 ≤ j < k ≤ r Here we have Υ q ( C i ) = � [ x i , x ϕ i ] , � x i � and [ C j , C ϕ k ][ C k , C ϕ j ] = � [ x j , x ϕ k ][ x k , x ϕ j ] , [ x j , x ϕ k ] � . In addition, ∆ q ( A ) = � r i ] � × � i = 1 � [ x i , x ϕ j < k � [ x j , x ϕ k ][ x k , x ϕ j ] � . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Write E q x i , [ x j , x ϕ X ( A ) := � � k ] | 1 ≤ i ≤ r , 1 ≤ j < k ≤ r � . We get Υ q ( A ) = ∆ q ( A ) E q X ( A ) . For q = 0 , we have ∆( A ) ∩ E X ( A ) = 1 , giving the direct decomposition: Υ( A ) = ∆( A ) × E X ( A ) . If q > 0 the above decomposition is not in general valid. In fact, Υ q ( G ) = T G and ∆ q ( G ) ≤ T , for any group G ; thus, such a decompositions will depend on the relationship between T and G . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Write E q x i , [ x j , x ϕ X ( A ) := � � k ] | 1 ≤ i ≤ r , 1 ≤ j < k ≤ r � . We get Υ q ( A ) = ∆ q ( A ) E q X ( A ) . For q = 0 , we have ∆( A ) ∩ E X ( A ) = 1 , giving the direct decomposition: Υ( A ) = ∆( A ) × E X ( A ) . If q > 0 the above decomposition is not in general valid. In fact, Υ q ( G ) = T G and ∆ q ( G ) ≤ T , for any group G ; thus, such a decompositions will depend on the relationship between T and G . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Write E q x i , [ x j , x ϕ X ( A ) := � � k ] | 1 ≤ i ≤ r , 1 ≤ j < k ≤ r � . We get Υ q ( A ) = ∆ q ( A ) E q X ( A ) . For q = 0 , we have ∆( A ) ∩ E X ( A ) = 1 , giving the direct decomposition: Υ( A ) = ∆( A ) × E X ( A ) . If q > 0 the above decomposition is not in general valid. In fact, Υ q ( G ) = T G and ∆ q ( G ) ≤ T , for any group G ; thus, such a decompositions will depend on the relationship between T and G . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Write E q x i , [ x j , x ϕ X ( A ) := � � k ] | 1 ≤ i ≤ r , 1 ≤ j < k ≤ r � . We get Υ q ( A ) = ∆ q ( A ) E q X ( A ) . For q = 0 , we have ∆( A ) ∩ E X ( A ) = 1 , giving the direct decomposition: Υ( A ) = ∆( A ) × E X ( A ) . If q > 0 the above decomposition is not in general valid. In fact, Υ q ( G ) = T G and ∆ q ( G ) ≤ T , for any group G ; thus, such a decompositions will depend on the relationship between T and G . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Write E q x i , [ x j , x ϕ X ( A ) := � � k ] | 1 ≤ i ≤ r , 1 ≤ j < k ≤ r � . We get Υ q ( A ) = ∆ q ( A ) E q X ( A ) . For q = 0 , we have ∆( A ) ∩ E X ( A ) = 1 , giving the direct decomposition: Υ( A ) = ∆( A ) × E X ( A ) . If q > 0 the above decomposition is not in general valid. In fact, Υ q ( G ) = T G and ∆ q ( G ) ≤ T , for any group G ; thus, such a decompositions will depend on the relationship between T and G . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Write E q x i , [ x j , x ϕ X ( A ) := � � k ] | 1 ≤ i ≤ r , 1 ≤ j < k ≤ r � . We get Υ q ( A ) = ∆ q ( A ) E q X ( A ) . For q = 0 , we have ∆( A ) ∩ E X ( A ) = 1 , giving the direct decomposition: Υ( A ) = ∆( A ) × E X ( A ) . If q > 0 the above decomposition is not in general valid. In fact, Υ q ( G ) = T G and ∆ q ( G ) ≤ T , for any group G ; thus, such a decompositions will depend on the relationship between T and G . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Remark on ∆ q ( G ) ∆ q ( G ) essentially depends on q and G ab = G / G ′ ∆ q ( G ) does not depend on the particular set of generators of G : [N.R., 1994] Let X = { x i } i ∈ I be a set of generators of grp G (assume I totally ordered). Then ∆ q ( G ) is generated by the set ∆ X = { s i := [ x i , x ϕ i ] , t jk := [ x j , x ϕ k ][ x k , x ϕ j ] , | i , j , k ∈ I , j < k } . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Remark on ∆ q ( G ) ∆ q ( G ) essentially depends on q and G ab = G / G ′ ∆ q ( G ) does not depend on the particular set of generators of G : [N.R., 1994] Let X = { x i } i ∈ I be a set of generators of grp G (assume I totally ordered). Then ∆ q ( G ) is generated by the set ∆ X = { s i := [ x i , x ϕ i ] , t jk := [ x j , x ϕ k ][ x k , x ϕ j ] , | i , j , k ∈ I , j < k } . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Remark on ∆ q ( G ) ∆ q ( G ) essentially depends on q and G ab = G / G ′ ∆ q ( G ) does not depend on the particular set of generators of G : [N.R., 1994] Let X = { x i } i ∈ I be a set of generators of grp G (assume I totally ordered). Then ∆ q ( G ) is generated by the set ∆ X = { s i := [ x i , x ϕ i ] , t jk := [ x j , x ϕ k ][ x k , x ϕ j ] , | i , j , k ∈ I , j < k } . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

For q = 0 , [Blyth, Fumagalli and Morigi, 2009] extending results of [Brown, Johnson & Robertson, 1987] and of [N.R., 1991, 1994] established conditions on f.g. G ab in order to describe the structure of G ⊗ G as a direct product of ∆( G ) and the exterior square G ∧ G : [BFM, 2009] Let G be a group s.t. G ab is f.g. If G ab has no element of order 2, or if G ′ has a complement in G , the G ⊗ G ∼ = ∆( G ) × ( G ∧ G ) We can extend above result in some way for q ≥ 1 and q odd If q = 2 then ∆ 2 ( G ) ≤ T ≤ G ; consequently, such a direct decomposition is not in general possible if q is even N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

For q = 0 , [Blyth, Fumagalli and Morigi, 2009] extending results of [Brown, Johnson & Robertson, 1987] and of [N.R., 1991, 1994] established conditions on f.g. G ab in order to describe the structure of G ⊗ G as a direct product of ∆( G ) and the exterior square G ∧ G : [BFM, 2009] Let G be a group s.t. G ab is f.g. If G ab has no element of order 2, or if G ′ has a complement in G , the G ⊗ G ∼ = ∆( G ) × ( G ∧ G ) We can extend above result in some way for q ≥ 1 and q odd If q = 2 then ∆ 2 ( G ) ≤ T ≤ G ; consequently, such a direct decomposition is not in general possible if q is even N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

For q = 0 , [Blyth, Fumagalli and Morigi, 2009] extending results of [Brown, Johnson & Robertson, 1987] and of [N.R., 1991, 1994] established conditions on f.g. G ab in order to describe the structure of G ⊗ G as a direct product of ∆( G ) and the exterior square G ∧ G : [BFM, 2009] Let G be a group s.t. G ab is f.g. If G ab has no element of order 2, or if G ′ has a complement in G , the G ⊗ G ∼ = ∆( G ) × ( G ∧ G ) We can extend above result in some way for q ≥ 1 and q odd If q = 2 then ∆ 2 ( G ) ≤ T ≤ G ; consequently, such a direct decomposition is not in general possible if q is even N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

For q = 0 , [Blyth, Fumagalli and Morigi, 2009] extending results of [Brown, Johnson & Robertson, 1987] and of [N.R., 1991, 1994] established conditions on f.g. G ab in order to describe the structure of G ⊗ G as a direct product of ∆( G ) and the exterior square G ∧ G : [BFM, 2009] Let G be a group s.t. G ab is f.g. If G ab has no element of order 2, or if G ′ has a complement in G , the G ⊗ G ∼ = ∆( G ) × ( G ∧ G ) We can extend above result in some way for q ≥ 1 and q odd If q = 2 then ∆ 2 ( G ) ≤ T ≤ G ; consequently, such a direct decomposition is not in general possible if q is even N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

[R & R, 2016] Let q > 1 be an odd integer and A a finitely generated abelian group given by the presentation A = � x 1 , . . . , x r | x n i i , [ x j , x k ] , 1 ≤ i , j , k ≤ r , j < k � , where we assume that n l + 1 = · · · = n r = 0 in case the free part of A is generated by { x l + 1 , . . . , x r } , 0 ≤ l ≤ r − 1. Write d i = gcd ( q , n i ) and d jk = gcd ( q , n j , n k ) , 1 ≤ i , j , k ≤ r , j < k . Then = � r i = 1 C d i × � ( i ) ∆ q ( A ) ∼ 1 ≤ j < k ≤ r C d jk and = A × � X ( A ) ∼ E q 1 ≤ j < k ≤ r C d jk ; ( ii ) Υ q ( A ) = ∆ q ( A ) × E q X ( A ) . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

[R & R, 2016] Let q > 1 be an odd integer and A a finitely generated abelian group given by the presentation A = � x 1 , . . . , x r | x n i i , [ x j , x k ] , 1 ≤ i , j , k ≤ r , j < k � , where we assume that n l + 1 = · · · = n r = 0 in case the free part of A is generated by { x l + 1 , . . . , x r } , 0 ≤ l ≤ r − 1. Write d i = gcd ( q , n i ) and d jk = gcd ( q , n j , n k ) , 1 ≤ i , j , k ≤ r , j < k . Then = � r i = 1 C d i × � ( i ) ∆ q ( A ) ∼ 1 ≤ j < k ≤ r C d jk and = A × � X ( A ) ∼ E q 1 ≤ j < k ≤ r C d jk ; ( ii ) Υ q ( A ) = ∆ q ( A ) × E q X ( A ) . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

[R & R, 2016] Let q > 1 be an odd integer and A a finitely generated abelian group given by the presentation A = � x 1 , . . . , x r | x n i i , [ x j , x k ] , 1 ≤ i , j , k ≤ r , j < k � , where we assume that n l + 1 = · · · = n r = 0 in case the free part of A is generated by { x l + 1 , . . . , x r } , 0 ≤ l ≤ r − 1. Write d i = gcd ( q , n i ) and d jk = gcd ( q , n j , n k ) , 1 ≤ i , j , k ≤ r , j < k . Then = � r i = 1 C d i × � ( i ) ∆ q ( A ) ∼ 1 ≤ j < k ≤ r C d jk and = A × � X ( A ) ∼ E q 1 ≤ j < k ≤ r C d jk ; ( ii ) Υ q ( A ) = ∆ q ( A ) × E q X ( A ) . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

[R & R, 2016] Let q > 1 be an odd integer and A a finitely generated abelian group given by the presentation A = � x 1 , . . . , x r | x n i i , [ x j , x k ] , 1 ≤ i , j , k ≤ r , j < k � , where we assume that n l + 1 = · · · = n r = 0 in case the free part of A is generated by { x l + 1 , . . . , x r } , 0 ≤ l ≤ r − 1. Write d i = gcd ( q , n i ) and d jk = gcd ( q , n j , n k ) , 1 ≤ i , j , k ≤ r , j < k . Then = � r i = 1 C d i × � ( i ) ∆ q ( A ) ∼ 1 ≤ j < k ≤ r C d jk and = A × � X ( A ) ∼ E q 1 ≤ j < k ≤ r C d jk ; ( ii ) Υ q ( A ) = ∆ q ( A ) × E q X ( A ) . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

G ab finiteley generated [Thm - R.& R. 2016] Let q > 1 be an odd integer and assume G ab is f.g. Then (i) ∆ q ( G ) ∩ E q ( G ) = 1; (ii) ∆ q ( G ) ∼ = ∆ q ( G ab ) ; = ∆ q ( G ab ) × ( G ∧ q G ) ; (iii) Υ q ( G ) ∼ (iv) If G ab is free abelian of rank r , then ∆ q ( G ) is a homocyclic � r + 1 � abelian group of exponent q , of rank . 2 N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

G ab finiteley generated [Thm - R.& R. 2016] Let q > 1 be an odd integer and assume G ab is f.g. Then (i) ∆ q ( G ) ∩ E q ( G ) = 1; (ii) ∆ q ( G ) ∼ = ∆ q ( G ab ) ; = ∆ q ( G ab ) × ( G ∧ q G ) ; (iii) Υ q ( G ) ∼ (iv) If G ab is free abelian of rank r , then ∆ q ( G ) is a homocyclic � r + 1 � abelian group of exponent q , of rank . 2 N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

[G. Ellis, 1989] - Computing G ∧ q G from a presentation of G Let q ≥ 0 and let F / R be a free presentation of G . Then G ∧ q G ∼ = F ′ F q / [ R , F ] R q . [Corollary] Let F n be the free group of rank n . Then (i) [RR, 2016] For q ≥ 1 and q odd, = C ( n + 1 2 ) F n ⊗ q F n ∼ × ( F n ) ′ ( F n ) q . q (ii) [BJR, 1987] For q = 0 , = C ( n + 1 2 ) F n ⊗ F n ∼ × ( F n ) ′ . ∞ N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

[G. Ellis, 1989] - Computing G ∧ q G from a presentation of G Let q ≥ 0 and let F / R be a free presentation of G . Then G ∧ q G ∼ = F ′ F q / [ R , F ] R q . [Corollary] Let F n be the free group of rank n . Then (i) [RR, 2016] For q ≥ 1 and q odd, = C ( n + 1 2 ) F n ⊗ q F n ∼ × ( F n ) ′ ( F n ) q . q (ii) [BJR, 1987] For q = 0 , = C ( n + 1 2 ) F n ⊗ F n ∼ × ( F n ) ′ . ∞ N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

[G. Ellis, 1989] - Computing G ∧ q G from a presentation of G Let q ≥ 0 and let F / R be a free presentation of G . Then G ∧ q G ∼ = F ′ F q / [ R , F ] R q . [Corollary] Let F n be the free group of rank n . Then (i) [RR, 2016] For q ≥ 1 and q odd, = C ( n + 1 2 ) F n ⊗ q F n ∼ × ( F n ) ′ ( F n ) q . q (ii) [BJR, 1987] For q = 0 , = C ( n + 1 2 ) F n ⊗ F n ∼ × ( F n ) ′ . ∞ N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

[G. Ellis, 1989] - Computing G ∧ q G from a presentation of G Let q ≥ 0 and let F / R be a free presentation of G . Then G ∧ q G ∼ = F ′ F q / [ R , F ] R q . [Corollary] Let F n be the free group of rank n . Then (i) [RR, 2016] For q ≥ 1 and q odd, = C ( n + 1 2 ) F n ⊗ q F n ∼ × ( F n ) ′ ( F n ) q . q (ii) [BJR, 1987] For q = 0 , = C ( n + 1 2 ) F n ⊗ F n ∼ × ( F n ) ′ . ∞ N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

[G. Ellis, 1989] - Computing G ∧ q G from a presentation of G Let q ≥ 0 and let F / R be a free presentation of G . Then G ∧ q G ∼ = F ′ F q / [ R , F ] R q . [Corollary] Let F n be the free group of rank n . Then (i) [RR, 2016] For q ≥ 1 and q odd, = C ( n + 1 2 ) F n ⊗ q F n ∼ × ( F n ) ′ ( F n ) q . q (ii) [BJR, 1987] For q = 0 , = C ( n + 1 2 ) F n ⊗ F n ∼ × ( F n ) ′ . ∞ N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

[G. Ellis, 1989] - Computing G ∧ q G from a presentation of G Let q ≥ 0 and let F / R be a free presentation of G . Then G ∧ q G ∼ = F ′ F q / [ R , F ] R q . [Corollary] Let F n be the free group of rank n . Then (i) [RR, 2016] For q ≥ 1 and q odd, = C ( n + 1 2 ) F n ⊗ q F n ∼ × ( F n ) ′ ( F n ) q . q (ii) [BJR, 1987] For q = 0 , = C ( n + 1 2 ) F n ⊗ F n ∼ × ( F n ) ′ . ∞ N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

[Corollary] Let N n , c = F n /γ c + 1 ( F n ) be the free nilpotent group of class c ≥ 1 and rank n > 1. Then (i) [R & R, 2016] For q ≥ 1 and q odd ( F n ) ′ ( F n ) q = C ( n + 1 2 ) N n , c ⊗ q N n , c ∼ × γ c + 1 ( F n ) q γ c + 2 ( F n ) . q (ii) [BFM 2010, Corollary 1.7] For q = 0 , = C ( n + 1 2 ) N n , c ⊗ N n , c ∼ × ( N n , c + 1 ) ′ . ∞ N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

[Corollary] Let N n , c = F n /γ c + 1 ( F n ) be the free nilpotent group of class c ≥ 1 and rank n > 1. Then (i) [R & R, 2016] For q ≥ 1 and q odd ( F n ) ′ ( F n ) q = C ( n + 1 2 ) N n , c ⊗ q N n , c ∼ × γ c + 1 ( F n ) q γ c + 2 ( F n ) . q (ii) [BFM 2010, Corollary 1.7] For q = 0 , = C ( n + 1 2 ) N n , c ⊗ N n , c ∼ × ( N n , c + 1 ) ′ . ∞ N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

[Corollary] Let N n , c = F n /γ c + 1 ( F n ) be the free nilpotent group of class c ≥ 1 and rank n > 1. Then (i) [R & R, 2016] For q ≥ 1 and q odd ( F n ) ′ ( F n ) q = C ( n + 1 2 ) N n , c ⊗ q N n , c ∼ × γ c + 1 ( F n ) q γ c + 2 ( F n ) . q (ii) [BFM 2010, Corollary 1.7] For q = 0 , = C ( n + 1 2 ) N n , c ⊗ N n , c ∼ × ( N n , c + 1 ) ′ . ∞ N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

f.g. nilpotent groups of class 2 [Thm] Let N n , 2 be the free nilpotent group of rank n > 1 and class 2, N n , 2 = F n /γ 3 ( F n ) . Then, (i) [M.Bacon, 1994] N n , 2 ⊗ N n , 2 is free abelian of rank 3 n ( n 2 + 3 n − 1 ) . 1 More precisely, we have N n , 2 ⊗ N n , 2 ∼ = ∆( F ab n ) × M ( N n , 2 ) × N ′ n , 2 . (ii) [R. R., 2016] For q > 1 and q odd, N n , 2 ⊗ q N n , 2 ∼ = ( C q ) (( ( n + 1 2 ) )+ M n ( 3 )) × N ′ n , 2 N q n , 2 , 3 ( n 3 − n ) is the q − rank of where M n ( 3 ) = 1 γ 3 ( N n , 2 ) /γ 3 ( N n , 2 ) q γ 4 ( N n , 2 ) . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

f.g. nilpotent groups of class 2 [Thm] Let N n , 2 be the free nilpotent group of rank n > 1 and class 2, N n , 2 = F n /γ 3 ( F n ) . Then, (i) [M.Bacon, 1994] N n , 2 ⊗ N n , 2 is free abelian of rank 3 n ( n 2 + 3 n − 1 ) . 1 More precisely, we have N n , 2 ⊗ N n , 2 ∼ = ∆( F ab n ) × M ( N n , 2 ) × N ′ n , 2 . (ii) [R. R., 2016] For q > 1 and q odd, N n , 2 ⊗ q N n , 2 ∼ = ( C q ) (( ( n + 1 2 ) )+ M n ( 3 )) × N ′ n , 2 N q n , 2 , 3 ( n 3 − n ) is the q − rank of where M n ( 3 ) = 1 γ 3 ( N n , 2 ) /γ 3 ( N n , 2 ) q γ 4 ( N n , 2 ) . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

f.g. nilpotent groups of class 2 [Thm] Let N n , 2 be the free nilpotent group of rank n > 1 and class 2, N n , 2 = F n /γ 3 ( F n ) . Then, (i) [M.Bacon, 1994] N n , 2 ⊗ N n , 2 is free abelian of rank 3 n ( n 2 + 3 n − 1 ) . 1 More precisely, we have N n , 2 ⊗ N n , 2 ∼ = ∆( F ab n ) × M ( N n , 2 ) × N ′ n , 2 . (ii) [R. R., 2016] For q > 1 and q odd, N n , 2 ⊗ q N n , 2 ∼ = ( C q ) (( ( n + 1 2 ) )+ M n ( 3 )) × N ′ n , 2 N q n , 2 , 3 ( n 3 − n ) is the q − rank of where M n ( 3 ) = 1 γ 3 ( N n , 2 ) /γ 3 ( N n , 2 ) q γ 4 ( N n , 2 ) . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

f.g. nilpotent groups of class 2 [Thm] Let N n , 2 be the free nilpotent group of rank n > 1 and class 2, N n , 2 = F n /γ 3 ( F n ) . Then, (i) [M.Bacon, 1994] N n , 2 ⊗ N n , 2 is free abelian of rank 3 n ( n 2 + 3 n − 1 ) . 1 More precisely, we have N n , 2 ⊗ N n , 2 ∼ = ∆( F ab n ) × M ( N n , 2 ) × N ′ n , 2 . (ii) [R. R., 2016] For q > 1 and q odd, N n , 2 ⊗ q N n , 2 ∼ = ( C q ) (( ( n + 1 2 ) )+ M n ( 3 )) × N ′ n , 2 N q n , 2 , 3 ( n 3 − n ) is the q − rank of where M n ( 3 ) = 1 γ 3 ( N n , 2 ) /γ 3 ( N n , 2 ) q γ 4 ( N n , 2 ) . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

f.g. nilpotent groups of class 2 [Thm] Let N n , 2 be the free nilpotent group of rank n > 1 and class 2, N n , 2 = F n /γ 3 ( F n ) . Then, (i) [M.Bacon, 1994] N n , 2 ⊗ N n , 2 is free abelian of rank 3 n ( n 2 + 3 n − 1 ) . 1 More precisely, we have N n , 2 ⊗ N n , 2 ∼ = ∆( F ab n ) × M ( N n , 2 ) × N ′ n , 2 . (ii) [R. R., 2016] For q > 1 and q odd, N n , 2 ⊗ q N n , 2 ∼ = ( C q ) (( ( n + 1 2 ) )+ M n ( 3 )) × N ′ n , 2 N q n , 2 , 3 ( n 3 − n ) is the q − rank of where M n ( 3 ) = 1 γ 3 ( N n , 2 ) /γ 3 ( N n , 2 ) q γ 4 ( N n , 2 ) . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

f.g. nilpotent groups of class 2 [Thm] Let N n , 2 be the free nilpotent group of rank n > 1 and class 2, N n , 2 = F n /γ 3 ( F n ) . Then, (i) [M.Bacon, 1994] N n , 2 ⊗ N n , 2 is free abelian of rank 3 n ( n 2 + 3 n − 1 ) . 1 More precisely, we have N n , 2 ⊗ N n , 2 ∼ = ∆( F ab n ) × M ( N n , 2 ) × N ′ n , 2 . (ii) [R. R., 2016] For q > 1 and q odd, N n , 2 ⊗ q N n , 2 ∼ = ( C q ) (( ( n + 1 2 ) )+ M n ( 3 )) × N ′ n , 2 N q n , 2 , 3 ( n 3 − n ) is the q − rank of where M n ( 3 ) = 1 γ 3 ( N n , 2 ) /γ 3 ( N n , 2 ) q γ 4 ( N n , 2 ) . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

f.g. nilpotent groups of class 2 [Thm] Let N n , 2 be the free nilpotent group of rank n > 1 and class 2, N n , 2 = F n /γ 3 ( F n ) . Then, (i) [M.Bacon, 1994] N n , 2 ⊗ N n , 2 is free abelian of rank 3 n ( n 2 + 3 n − 1 ) . 1 More precisely, we have N n , 2 ⊗ N n , 2 ∼ = ∆( F ab n ) × M ( N n , 2 ) × N ′ n , 2 . (ii) [R. R., 2016] For q > 1 and q odd, N n , 2 ⊗ q N n , 2 ∼ = ( C q ) (( ( n + 1 2 ) )+ M n ( 3 )) × N ′ n , 2 N q n , 2 , 3 ( n 3 − n ) is the q − rank of where M n ( 3 ) = 1 γ 3 ( N n , 2 ) /γ 3 ( N n , 2 ) q γ 4 ( N n , 2 ) . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Consequently, for q > 1 and q odd, d ( N n , 2 ⊗ q N n , 2 ) = 1 3 ( n 3 + 3 n 2 + 2 n ) . This is the least upper bound for d ( G ⊗ q G ) , G a class 2 nilpotent group with d ( G ) = n : [E.Rodrigues, 2011] Let G be a nilpotent group of class 2 with d ( G ) = n . Then G ⊗ q G can be generated by at most 1 3 n ( n 2 + 3 n + 2 ) elements. In particular, if G is finite and gcd ( q , | G | ) = 1 then d ( G ⊗ q G ) ≤ n 2 . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Consequently, for q > 1 and q odd, d ( N n , 2 ⊗ q N n , 2 ) = 1 3 ( n 3 + 3 n 2 + 2 n ) . This is the least upper bound for d ( G ⊗ q G ) , G a class 2 nilpotent group with d ( G ) = n : [E.Rodrigues, 2011] Let G be a nilpotent group of class 2 with d ( G ) = n . Then G ⊗ q G can be generated by at most 1 3 n ( n 2 + 3 n + 2 ) elements. In particular, if G is finite and gcd ( q , | G | ) = 1 then d ( G ⊗ q G ) ≤ n 2 . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Consequently, for q > 1 and q odd, d ( N n , 2 ⊗ q N n , 2 ) = 1 3 ( n 3 + 3 n 2 + 2 n ) . This is the least upper bound for d ( G ⊗ q G ) , G a class 2 nilpotent group with d ( G ) = n : [E.Rodrigues, 2011] Let G be a nilpotent group of class 2 with d ( G ) = n . Then G ⊗ q G can be generated by at most 1 3 n ( n 2 + 3 n + 2 ) elements. In particular, if G is finite and gcd ( q , | G | ) = 1 then d ( G ⊗ q G ) ≤ n 2 . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Consequently, for q > 1 and q odd, d ( N n , 2 ⊗ q N n , 2 ) = 1 3 ( n 3 + 3 n 2 + 2 n ) . This is the least upper bound for d ( G ⊗ q G ) , G a class 2 nilpotent group with d ( G ) = n : [E.Rodrigues, 2011] Let G be a nilpotent group of class 2 with d ( G ) = n . Then G ⊗ q G can be generated by at most 1 3 n ( n 2 + 3 n + 2 ) elements. In particular, if G is finite and gcd ( q , | G | ) = 1 then d ( G ⊗ q G ) ≤ n 2 . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

Consequently, for q > 1 and q odd, d ( N n , 2 ⊗ q N n , 2 ) = 1 3 ( n 3 + 3 n 2 + 2 n ) . This is the least upper bound for d ( G ⊗ q G ) , G a class 2 nilpotent group with d ( G ) = n : [E.Rodrigues, 2011] Let G be a nilpotent group of class 2 with d ( G ) = n . Then G ⊗ q G can be generated by at most 1 3 n ( n 2 + 3 n + 2 ) elements. In particular, if G is finite and gcd ( q , | G | ) = 1 then d ( G ⊗ q G ) ≤ n 2 . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

An example: The Heisenberg group in detail Let H = F 2 /γ 3 ( F 2 ) be the Heisenberg group, where F 2 denotes the free group of rank 2. By previous thm, part (ii) we have, for q > 1, q odd: H ⊗ q H ∼ = ( C q ) 5 × H ′ H q Now, H has the polycyclic presentation H = � x , y , z | [ y , x ] = z , [ z , x ] = 1 = [ z , y ] � . (7) and thus Υ q ( H ) is generated by { [ x , x ϕ ] , [ x , y ϕ ] , [ y , x ϕ ] , [ y , y ϕ ] , [ x , z ϕ ] , [ y , z ϕ ] , ˆ x , ˆ y } . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

An example: The Heisenberg group in detail Let H = F 2 /γ 3 ( F 2 ) be the Heisenberg group, where F 2 denotes the free group of rank 2. By previous thm, part (ii) we have, for q > 1, q odd: H ⊗ q H ∼ = ( C q ) 5 × H ′ H q Now, H has the polycyclic presentation H = � x , y , z | [ y , x ] = z , [ z , x ] = 1 = [ z , y ] � . (7) and thus Υ q ( H ) is generated by { [ x , x ϕ ] , [ x , y ϕ ] , [ y , x ϕ ] , [ y , y ϕ ] , [ x , z ϕ ] , [ y , z ϕ ] , ˆ x , ˆ y } . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

An example: The Heisenberg group in detail Let H = F 2 /γ 3 ( F 2 ) be the Heisenberg group, where F 2 denotes the free group of rank 2. By previous thm, part (ii) we have, for q > 1, q odd: H ⊗ q H ∼ = ( C q ) 5 × H ′ H q Now, H has the polycyclic presentation H = � x , y , z | [ y , x ] = z , [ z , x ] = 1 = [ z , y ] � . (7) and thus Υ q ( H ) is generated by { [ x , x ϕ ] , [ x , y ϕ ] , [ y , x ϕ ] , [ y , y ϕ ] , [ x , z ϕ ] , [ y , z ϕ ] , ˆ x , ˆ y } . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

An example: The Heisenberg group in detail Let H = F 2 /γ 3 ( F 2 ) be the Heisenberg group, where F 2 denotes the free group of rank 2. By previous thm, part (ii) we have, for q > 1, q odd: H ⊗ q H ∼ = ( C q ) 5 × H ′ H q Now, H has the polycyclic presentation H = � x , y , z | [ y , x ] = z , [ z , x ] = 1 = [ z , y ] � . (7) and thus Υ q ( H ) is generated by { [ x , x ϕ ] , [ x , y ϕ ] , [ y , x ϕ ] , [ y , y ϕ ] , [ x , z ϕ ] , [ y , z ϕ ] , ˆ x , ˆ y } . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

The Heisenberg grp, cont. In addition, the following relations hold in ν q ( H ) : [ x , x ϕ ] q = [ x , z ϕ ] q = [ y , z ϕ ] q = [ y , y ϕ ] q = 1 ([ x , y ϕ ][ y , x ϕ ]) q = 1 [ x , y ϕ ] q = � z ) − 1 = [ y , x ϕ ] − q z − 1 = ( � x ] = [ y , x ϕ ] q 2 (= � z q ) , [ � y , � and all other generators commute. ∴ Υ q ( H ) is a homomorphic image of the y , [ y , x ϕ ] , [ x , z ϕ ] , [ y , z ϕ ] , [ x , y ϕ ][ y , x ϕ ] , [ x , x ϕ ] , [ y , y ϕ ] | � ˆ x , ˆ above relations � . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

The Heisenberg grp, cont. In addition, the following relations hold in ν q ( H ) : [ x , x ϕ ] q = [ x , z ϕ ] q = [ y , z ϕ ] q = [ y , y ϕ ] q = 1 ([ x , y ϕ ][ y , x ϕ ]) q = 1 [ x , y ϕ ] q = � z ) − 1 = [ y , x ϕ ] − q z − 1 = ( � x ] = [ y , x ϕ ] q 2 (= � z q ) , [ � y , � and all other generators commute. ∴ Υ q ( H ) is a homomorphic image of the y , [ y , x ϕ ] , [ x , z ϕ ] , [ y , z ϕ ] , [ x , y ϕ ][ y , x ϕ ] , [ x , x ϕ ] , [ y , y ϕ ] | � ˆ x , ˆ above relations � . N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

The Heisenberg grp, cont. We have: = H ′ H q = � x q , y q , z � ≤ H ; y , [ y , x ϕ ] � ∼ H 1 = � ˆ x , ˆ By our previous results, if q = 0 or if q ≥ 1 and q is odd, then = ( C q ) 3 ∼ ∆ q ( H ) = � [ x , y ϕ ][ y , x ϕ ] , [ x , x ϕ ] , [ y , y ϕ ] � ∼ = ∆ q ( G ab ) H ∧ q H ∼ y , [ y , x ϕ ] , [ x , z ϕ ] , [ y , z ϕ ] � ∼ = E q ( G ) = � ˆ = H 1 × ( C q ) 2 x , ˆ Finally we have H ⊗ q H ∼ = ∆ q ( H ab ) × H ∧ q H ∼ = H 1 × ( C q ) 5 , where H 1 ∼ = � a , b , c | [ b , a ] = c q 2 , [ c , a ] = 1 = [ c , b ] � ∼ = H ′ H q . In particular, for q = 0 we find H ⊗ H ∼ = ( C ∞ ) 6 , which is a result of R.Aboughazi, 1987. N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

The Heisenberg grp, cont. We have: = H ′ H q = � x q , y q , z � ≤ H ; y , [ y , x ϕ ] � ∼ H 1 = � ˆ x , ˆ By our previous results, if q = 0 or if q ≥ 1 and q is odd, then = ( C q ) 3 ∼ ∆ q ( H ) = � [ x , y ϕ ][ y , x ϕ ] , [ x , x ϕ ] , [ y , y ϕ ] � ∼ = ∆ q ( G ab ) H ∧ q H ∼ y , [ y , x ϕ ] , [ x , z ϕ ] , [ y , z ϕ ] � ∼ = E q ( G ) = � ˆ = H 1 × ( C q ) 2 x , ˆ Finally we have H ⊗ q H ∼ = ∆ q ( H ab ) × H ∧ q H ∼ = H 1 × ( C q ) 5 , where H 1 ∼ = � a , b , c | [ b , a ] = c q 2 , [ c , a ] = 1 = [ c , b ] � ∼ = H ′ H q . In particular, for q = 0 we find H ⊗ H ∼ = ( C ∞ ) 6 , which is a result of R.Aboughazi, 1987. N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

The Heisenberg grp, cont. We have: = H ′ H q = � x q , y q , z � ≤ H ; y , [ y , x ϕ ] � ∼ H 1 = � ˆ x , ˆ By our previous results, if q = 0 or if q ≥ 1 and q is odd, then = ( C q ) 3 ∼ ∆ q ( H ) = � [ x , y ϕ ][ y , x ϕ ] , [ x , x ϕ ] , [ y , y ϕ ] � ∼ = ∆ q ( G ab ) H ∧ q H ∼ y , [ y , x ϕ ] , [ x , z ϕ ] , [ y , z ϕ ] � ∼ = E q ( G ) = � ˆ = H 1 × ( C q ) 2 x , ˆ Finally we have H ⊗ q H ∼ = ∆ q ( H ab ) × H ∧ q H ∼ = H 1 × ( C q ) 5 , where H 1 ∼ = � a , b , c | [ b , a ] = c q 2 , [ c , a ] = 1 = [ c , b ] � ∼ = H ′ H q . In particular, for q = 0 we find H ⊗ H ∼ = ( C ∞ ) 6 , which is a result of R.Aboughazi, 1987. N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

The Heisenberg grp, cont. We have: = H ′ H q = � x q , y q , z � ≤ H ; y , [ y , x ϕ ] � ∼ H 1 = � ˆ x , ˆ By our previous results, if q = 0 or if q ≥ 1 and q is odd, then = ( C q ) 3 ∼ ∆ q ( H ) = � [ x , y ϕ ][ y , x ϕ ] , [ x , x ϕ ] , [ y , y ϕ ] � ∼ = ∆ q ( G ab ) H ∧ q H ∼ y , [ y , x ϕ ] , [ x , z ϕ ] , [ y , z ϕ ] � ∼ = E q ( G ) = � ˆ = H 1 × ( C q ) 2 x , ˆ Finally we have H ⊗ q H ∼ = ∆ q ( H ab ) × H ∧ q H ∼ = H 1 × ( C q ) 5 , where H 1 ∼ = � a , b , c | [ b , a ] = c q 2 , [ c , a ] = 1 = [ c , b ] � ∼ = H ′ H q . In particular, for q = 0 we find H ⊗ H ∼ = ( C ∞ ) 6 , which is a result of R.Aboughazi, 1987. N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

The Heisenberg grp, cont. We have: = H ′ H q = � x q , y q , z � ≤ H ; y , [ y , x ϕ ] � ∼ H 1 = � ˆ x , ˆ By our previous results, if q = 0 or if q ≥ 1 and q is odd, then = ( C q ) 3 ∼ ∆ q ( H ) = � [ x , y ϕ ][ y , x ϕ ] , [ x , x ϕ ] , [ y , y ϕ ] � ∼ = ∆ q ( G ab ) H ∧ q H ∼ y , [ y , x ϕ ] , [ x , z ϕ ] , [ y , z ϕ ] � ∼ = E q ( G ) = � ˆ = H 1 × ( C q ) 2 x , ˆ Finally we have H ⊗ q H ∼ = ∆ q ( H ab ) × H ∧ q H ∼ = H 1 × ( C q ) 5 , where H 1 ∼ = � a , b , c | [ b , a ] = c q 2 , [ c , a ] = 1 = [ c , b ] � ∼ = H ′ H q . In particular, for q = 0 we find H ⊗ H ∼ = ( C ∞ ) 6 , which is a result of R.Aboughazi, 1987. N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

The Heisenberg grp, cont. We have: = H ′ H q = � x q , y q , z � ≤ H ; y , [ y , x ϕ ] � ∼ H 1 = � ˆ x , ˆ By our previous results, if q = 0 or if q ≥ 1 and q is odd, then = ( C q ) 3 ∼ ∆ q ( H ) = � [ x , y ϕ ][ y , x ϕ ] , [ x , x ϕ ] , [ y , y ϕ ] � ∼ = ∆ q ( G ab ) H ∧ q H ∼ y , [ y , x ϕ ] , [ x , z ϕ ] , [ y , z ϕ ] � ∼ = E q ( G ) = � ˆ = H 1 × ( C q ) 2 x , ˆ Finally we have H ⊗ q H ∼ = ∆ q ( H ab ) × H ∧ q H ∼ = H 1 × ( C q ) 5 , where H 1 ∼ = � a , b , c | [ b , a ] = c q 2 , [ c , a ] = 1 = [ c , b ] � ∼ = H ′ H q . In particular, for q = 0 we find H ⊗ H ∼ = ( C ∞ ) 6 , which is a result of R.Aboughazi, 1987. N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

The Heisenberg grp, cont. We have: = H ′ H q = � x q , y q , z � ≤ H ; y , [ y , x ϕ ] � ∼ H 1 = � ˆ x , ˆ By our previous results, if q = 0 or if q ≥ 1 and q is odd, then = ( C q ) 3 ∼ ∆ q ( H ) = � [ x , y ϕ ][ y , x ϕ ] , [ x , x ϕ ] , [ y , y ϕ ] � ∼ = ∆ q ( G ab ) H ∧ q H ∼ y , [ y , x ϕ ] , [ x , z ϕ ] , [ y , z ϕ ] � ∼ = E q ( G ) = � ˆ = H 1 × ( C q ) 2 x , ˆ Finally we have H ⊗ q H ∼ = ∆ q ( H ab ) × H ∧ q H ∼ = H 1 × ( C q ) 5 , where H 1 ∼ = � a , b , c | [ b , a ] = c q 2 , [ c , a ] = 1 = [ c , b ] � ∼ = H ′ H q . In particular, for q = 0 we find H ⊗ H ∼ = ( C ∞ ) 6 , which is a result of R.Aboughazi, 1987. N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

The Heisenberg grp, cont. We have: = H ′ H q = � x q , y q , z � ≤ H ; y , [ y , x ϕ ] � ∼ H 1 = � ˆ x , ˆ By our previous results, if q = 0 or if q ≥ 1 and q is odd, then = ( C q ) 3 ∼ ∆ q ( H ) = � [ x , y ϕ ][ y , x ϕ ] , [ x , x ϕ ] , [ y , y ϕ ] � ∼ = ∆ q ( G ab ) H ∧ q H ∼ y , [ y , x ϕ ] , [ x , z ϕ ] , [ y , z ϕ ] � ∼ = E q ( G ) = � ˆ = H 1 × ( C q ) 2 x , ˆ Finally we have H ⊗ q H ∼ = ∆ q ( H ab ) × H ∧ q H ∼ = H 1 × ( C q ) 5 , where H 1 ∼ = � a , b , c | [ b , a ] = c q 2 , [ c , a ] = 1 = [ c , b ] � ∼ = H ′ H q . In particular, for q = 0 we find H ⊗ H ∼ = ( C ∞ ) 6 , which is a result of R.Aboughazi, 1987. N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

The Heisenberg grp, cont. We have: = H ′ H q = � x q , y q , z � ≤ H ; y , [ y , x ϕ ] � ∼ H 1 = � ˆ x , ˆ By our previous results, if q = 0 or if q ≥ 1 and q is odd, then = ( C q ) 3 ∼ ∆ q ( H ) = � [ x , y ϕ ][ y , x ϕ ] , [ x , x ϕ ] , [ y , y ϕ ] � ∼ = ∆ q ( G ab ) H ∧ q H ∼ y , [ y , x ϕ ] , [ x , z ϕ ] , [ y , z ϕ ] � ∼ = E q ( G ) = � ˆ = H 1 × ( C q ) 2 x , ˆ Finally we have H ⊗ q H ∼ = ∆ q ( H ab ) × H ∧ q H ∼ = H 1 × ( C q ) 5 , where H 1 ∼ = � a , b , c | [ b , a ] = c q 2 , [ c , a ] = 1 = [ c , b ] � ∼ = H ′ H q . In particular, for q = 0 we find H ⊗ H ∼ = ( C ∞ ) 6 , which is a result of R.Aboughazi, 1987. N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

The Heisenberg grp, cont. We have: = H ′ H q = � x q , y q , z � ≤ H ; y , [ y , x ϕ ] � ∼ H 1 = � ˆ x , ˆ By our previous results, if q = 0 or if q ≥ 1 and q is odd, then = ( C q ) 3 ∼ ∆ q ( H ) = � [ x , y ϕ ][ y , x ϕ ] , [ x , x ϕ ] , [ y , y ϕ ] � ∼ = ∆ q ( G ab ) H ∧ q H ∼ y , [ y , x ϕ ] , [ x , z ϕ ] , [ y , z ϕ ] � ∼ = E q ( G ) = � ˆ = H 1 × ( C q ) 2 x , ˆ Finally we have H ⊗ q H ∼ = ∆ q ( H ab ) × H ∧ q H ∼ = H 1 × ( C q ) 5 , where H 1 ∼ = � a , b , c | [ b , a ] = c q 2 , [ c , a ] = 1 = [ c , b ] � ∼ = H ′ H q . In particular, for q = 0 we find H ⊗ H ∼ = ( C ∞ ) 6 , which is a result of R.Aboughazi, 1987. N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

G polycyclic Let G be polycyclic, given by a consistent polycyclic presentation, say G = F / R Our procedure is an adaptation to all q ≥ 0 of a method described by Eick and Nickel (2008) for the case q = 0 We can find a consistent polycyclic presentation for F G ∗ = R q [ R , F ] From this we get a consistent pcp for G ∧ G ∼ = ( G ∗ ) ′ ( G ∗ ) q N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

G polycyclic Let G be polycyclic, given by a consistent polycyclic presentation, say G = F / R Our procedure is an adaptation to all q ≥ 0 of a method described by Eick and Nickel (2008) for the case q = 0 We can find a consistent polycyclic presentation for F G ∗ = R q [ R , F ] From this we get a consistent pcp for G ∧ G ∼ = ( G ∗ ) ′ ( G ∗ ) q N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

G polycyclic Let G be polycyclic, given by a consistent polycyclic presentation, say G = F / R Our procedure is an adaptation to all q ≥ 0 of a method described by Eick and Nickel (2008) for the case q = 0 We can find a consistent polycyclic presentation for F G ∗ = R q [ R , F ] From this we get a consistent pcp for G ∧ G ∼ = ( G ∗ ) ′ ( G ∗ ) q N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

G polycyclic Let G be polycyclic, given by a consistent polycyclic presentation, say G = F / R Our procedure is an adaptation to all q ≥ 0 of a method described by Eick and Nickel (2008) for the case q = 0 We can find a consistent polycyclic presentation for F G ∗ = R q [ R , F ] From this we get a consistent pcp for G ∧ G ∼ = ( G ∗ ) ′ ( G ∗ ) q N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

G polycyclic Let G be polycyclic, given by a consistent polycyclic presentation, say G = F / R Our procedure is an adaptation to all q ≥ 0 of a method described by Eick and Nickel (2008) for the case q = 0 We can find a consistent polycyclic presentation for F G ∗ = R q [ R , F ] From this we get a consistent pcp for G ∧ G ∼ = ( G ∗ ) ′ ( G ∗ ) q N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

A pcp for τ q ( G ) Let τ q ( G ) := ν q ( G ) / ∆ q ( G ) . We have τ q ( G ) ∼ = ( G ∧ q G ) ⋊ ( G × G ) . Can find a consistent pcp for τ q ( G ) . Need the concept of a q − biderivation, which extends the concept of a crossed pairing (biderivation) to the context of q-tensor squares Crossed pairings have been used in order to determine homomorphic images of the non-abelian tensor square G ⊗ G . Let G and L be arbitrary groups. N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

A pcp for τ q ( G ) Let τ q ( G ) := ν q ( G ) / ∆ q ( G ) . We have τ q ( G ) ∼ = ( G ∧ q G ) ⋊ ( G × G ) . Can find a consistent pcp for τ q ( G ) . Need the concept of a q − biderivation, which extends the concept of a crossed pairing (biderivation) to the context of q-tensor squares Crossed pairings have been used in order to determine homomorphic images of the non-abelian tensor square G ⊗ G . Let G and L be arbitrary groups. N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017

A pcp for τ q ( G ) Let τ q ( G ) := ν q ( G ) / ∆ q ( G ) . We have τ q ( G ) ∼ = ( G ∧ q G ) ⋊ ( G × G ) . Can find a consistent pcp for τ q ( G ) . Need the concept of a q − biderivation, which extends the concept of a crossed pairing (biderivation) to the context of q-tensor squares Crossed pairings have been used in order to determine homomorphic images of the non-abelian tensor square G ⊗ G . Let G and L be arbitrary groups. N.R.Rocco - UnB, joint with E.Rodrigues and I.Dias Groups St Andrews 2017 - Birmingham, UK - 05–13 Aug 2017