Normal Complement Problem for Metacyclic Groups Surinder Kaur - PowerPoint PPT Presentation

Normal Complement Problem for Metacyclic Groups Surinder Kaur (Joint work with Manju Khan) Indian Institute of Technology Ropar email: surinder.kaur@iitrpr.ac.in Groups, Rings and Associated Structures June 14, 2019 iitrpr.pdf

Notations G - a group iitrpr.pdf

Notations G - a group F - a field iitrpr.pdf

Notations G - a group F - a field FG - group algebra of G over F iitrpr.pdf

Notations G - a group F - a field FG - group algebra of G over F U ( FG ) - unit group of FG iitrpr.pdf

Notations G - a group F - a field FG - group algebra of G over F U ( FG ) - unit group of FG V ( FG ) - normalized unit group of FG iitrpr.pdf

Notations G - a group F - a field FG - group algebra of G over F U ( FG ) - unit group of FG V ( FG ) - normalized unit group of FG Γ( H ) - two-sided ideal generated by elements ( h − 1) in FG iitrpr.pdf

Normal Complement Problem Normal Complement Problem (R. K. Dennis* (1977)) For what groups G and rings R there exists a homomorphism φ : U ( RG ) → G such that φ is split by the natural inclusion i : G → U ( RG )? * R. Keith Dennis, The structure of the unit group of group rings, Lecture Notes in Pure and Appl. Math., iitrpr.pdf Vol. 26, 1977

Results known by far Let L p denote the set of finite p -groups G which have a normal complement in V ( F p G ) . Then iitrpr.pdf * L. E. Moran and R. N. Tench, Normal complements in mod p-envelopes. Israel J. Math., Israel J. Math.

Results known by far Let L p denote the set of finite p -groups G which have a normal complement in V ( F p G ) . Then Moran and Trench (1977)* Finite ableian p -groups lie in L p . 1 iitrpr.pdf * L. E. Moran and R. N. Tench, Normal complements in mod p-envelopes. Israel J. Math., Israel J. Math.

Results known by far Let L p denote the set of finite p -groups G which have a normal complement in V ( F p G ) . Then Moran and Trench (1977)* Finite ableian p -groups lie in L p . 1 D 8 and Q 8 belong to L 2 . 2 iitrpr.pdf * L. E. Moran and R. N. Tench, Normal complements in mod p-envelopes. Israel J. Math., Israel J. Math.

Results known by far Let L p denote the set of finite p -groups G which have a normal complement in V ( F p G ) . Then Moran and Trench (1977)* Finite ableian p -groups lie in L p . 1 D 8 and Q 8 belong to L 2 . 2 Finite p -groups of exponent p and nilpotency class 2 lie in L p . 3 iitrpr.pdf * L. E. Moran and R. N. Tench, Normal complements in mod p-envelopes. Israel J. Math., Israel J. Math.

Results known by far Let L p denote the set of finite p -groups G which have a normal complement in V ( F p G ) . Then Moran and Trench (1977)* Finite ableian p -groups lie in L p . 1 D 8 and Q 8 belong to L 2 . 2 Finite p -groups of exponent p and nilpotency class 2 lie in L p . 3 D 16 / ∈ L 2 . 4 iitrpr.pdf * L. E. Moran and R. N. Tench, Normal complements in mod p-envelopes. Israel J. Math., Israel J. Math.

L. Ivory (1980)** The dihedral, semi-dihedral and quaternion groups of order 16 and greater 1 do not lie in L 2 . iitrpr.pdf ** Ivory, Lee R., A note on normal complements in mod p envelopes, Proc. Amer. Math. Soc.

L. Ivory (1980)** The dihedral, semi-dihedral and quaternion groups of order 16 and greater 1 do not lie in L 2 . 11 of the 14 groups of order 16 are in L 2 . 2 iitrpr.pdf ** Ivory, Lee R., A note on normal complements in mod p envelopes, Proc. Amer. Math. Soc.

Normal complement problem for some metacyclic p -groups Theorem* (Kaur and Khan) Let G be a finite split metacyclic p -group with nilpotency class 2 and F be the field with p elements. Then G has a normal complement in V ( FG ) . * S. Kaur, M. Khan, A note on Normal Complement Problem for Split Metacyclic Groups. Communications in iitrpr.pdf Algebra (2019)

Sketch of proof: Let G = A ⋊ B , where A = � a � and B = � b � are cyclic groups of order p n and p m respectively. Then V ( FG ) = (1 + Γ( A )) ⋊ V ( FB ) . iitrpr.pdf * R. Sandling, Units in the modular group algebra of a finite abelian p-group, J. Pure Appl. Algebra (1984).

Sketch of proof: Let G = A ⋊ B , where A = � a � and B = � b � are cyclic groups of order p n and p m respectively. Then V ( FG ) = (1 + Γ( A )) ⋊ V ( FB ) . p m − 1 � � (1 + ( b − 1) i ) � . Write V ( FB ) = M × � b � . It is known* that V ( FB ) = i =1 i � = rp iitrpr.pdf * R. Sandling, Units in the modular group algebra of a finite abelian p-group, J. Pure Appl. Algebra (1984).

Sketch of proof: Let G = A ⋊ B , where A = � a � and B = � b � are cyclic groups of order p n and p m respectively. Then V ( FG ) = (1 + Γ( A )) ⋊ V ( FB ) . p m − 1 � � (1 + ( b − 1) i ) � . Write V ( FB ) = M × � b � . It is known* that V ( FB ) = i =1 i � = rp p n − 1 α a i ( a i − 1) ∈ (1 + Γ( A )) , where α a i ∈ FB . Define � Let x = 1 + i =1 φ : (1 + Γ( A )) → A by the rule � pn − 1 i ǫ ( α ai ) . x �→ a i =1 It can be checked that φ is an epimorphism. iitrpr.pdf * R. Sandling, Units in the modular group algebra of a finite abelian p-group, J. Pure Appl. Algebra (1984).

Sketch of proof: Let G = A ⋊ B , where A = � a � and B = � b � are cyclic groups of order p n and p m respectively. Then V ( FG ) = (1 + Γ( A )) ⋊ V ( FB ) . p m − 1 � � (1 + ( b − 1) i ) � . Write V ( FB ) = M × � b � . It is known* that V ( FB ) = i =1 i � = rp p n − 1 α a i ( a i − 1) ∈ (1 + Γ( A )) , where α a i ∈ FB . Define � Let x = 1 + i =1 φ : (1 + Γ( A )) → A by the rule � pn − 1 i ǫ ( α ai ) . x �→ a i =1 It can be checked that φ is an epimorphism. Thus (1 + Γ( A )) = ker ( φ ) ⋊ A . iitrpr.pdf * R. Sandling, Units in the modular group algebra of a finite abelian p-group, J. Pure Appl. Algebra (1984).

Sketch of proof: Let G = A ⋊ B , where A = � a � and B = � b � are cyclic groups of order p n and p m respectively. Then V ( FG ) = (1 + Γ( A )) ⋊ V ( FB ) . p m − 1 � � (1 + ( b − 1) i ) � . Write V ( FB ) = M × � b � . It is known* that V ( FB ) = i =1 i � = rp p n − 1 α a i ( a i − 1) ∈ (1 + Γ( A )) , where α a i ∈ FB . Define � Let x = 1 + i =1 φ : (1 + Γ( A )) → A by the rule � pn − 1 i ǫ ( α ai ) . x �→ a i =1 It can be checked that φ is an epimorphism. Thus (1 + Γ( A )) = ker ( φ ) ⋊ A . Finally, we obtain that V ( FG ) = ( ker ( φ ) ⋊ M ) ⋊ ( � a � ⋊ � b � ) . iitrpr.pdf * R. Sandling, Units in the modular group algebra of a finite abelian p-group, J. Pure Appl. Algebra (1984).

Normal complement of D 2 . 3 m in V ( FD 2 . 3 m ) Let p be an odd prime and F be a finite field of characteristic p . Let D 2 p m = � a , b | a p m = b 2 = e , b − 1 ab = a − 1 � be the dihedral group of order 2 p m . If A is the normal subgroup of D 2 p m , then V ( FD 2 p m ) = (1 + Γ( A )) ⋊ V ( F ( � b � )) . iitrpr.pdf

Normal complement of D 2 . 3 m in V ( FD 2 . 3 m ) Let p be an odd prime and F be a finite field of characteristic p . Let D 2 p m = � a , b | a p m = b 2 = e , b − 1 ab = a − 1 � be the dihedral group of order 2 p m . If A is the normal subgroup of D 2 p m , then V ( FD 2 p m ) = (1 + Γ( A )) ⋊ V ( F ( � b � )) . Theorem Let F be the field with 3 elements. Then the dihedral group D 2 . 3 m has a normal complement in the normalized unit group V ( FD 2 . 3 m ) . iitrpr.pdf

Proof: Consider the epimorphism φ : (1 + Γ( A )) → A defined by the rule p m − 1 � pm − 1 ( α i ( a i − 1) + β i ( a i − 1) b ) �→ a � i ( α i + β i ) . 1 + i =1 i =1 Thus (1 + Γ( A )) = ker ( φ ) ⋊ A . Since � b � normalizes ker ( φ ) , we have V ( FD 2 . 3 m ) = ker ( φ ) ⋊ D 2 . 3 m . iitrpr.pdf

Proof: Consider the epimorphism φ : (1 + Γ( A )) → A defined by the rule p m − 1 � pm − 1 ( α i ( a i − 1) + β i ( a i − 1) b ) �→ a � i ( α i + β i ) . 1 + i =1 i =1 Thus (1 + Γ( A )) = ker ( φ ) ⋊ A . Since � b � normalizes ker ( φ ) , we have V ( FD 2 . 3 m ) = ker ( φ ) ⋊ D 2 . 3 m . Corollary Let FS 3 be the group algebra of symmetric group S 3 over the field F with 3 elements. Then the elementary abelian 3 -group N = � (1 + ( a − 1) + 2( a − 1) b ) � × Z ( V ( FS 3 )) is a normal complement of S 3 in V ( FS 3 ) . iitrpr.pdf

A Partition of V ( FD 2 p m ) \ (1 + Γ( A )) Let V ( F ( � b � )) = � γ � , where γ = α + β b , for some α, β ∈ F . Define the set � � γ i z | 1 ≤ i ≤ | F ∗ | , z ∈ C (1+Γ( A )) ( b ) S = . iitrpr.pdf

A Partition of V ( FD 2 p m ) \ (1 + Γ( A )) Let V ( F ( � b � )) = � γ � , where γ = α + β b , for some α, β ∈ F . Define the set � � γ i z | 1 ≤ i ≤ | F ∗ | , z ∈ C (1+Γ( A )) ( b ) S = . Theorem � V ( FD 2 p m ) \ (1 + Γ( A )) = Cl ( s ) , where Cl ( s ) denotes the conjugacy class of s ∈ S s in V ( FD 2 p m ) . iitrpr.pdf

Normal complement problem for FD 2 p m , when | F | > 3 Theorem Let F be a finite field of characteristic p , where p is an odd prime. Then D 2 p m does not have normal complement in V ( FD 2 p m ) , for | F | > 3 . iitrpr.pdf

Sketch of Proof: Suppose there exists an epimorphism φ : V ( FD 2 p m ) → D 2 p m fixing elements of D 2 p m such that V ( FD 2 p m ) = N ⋊ D 2 p m , where N = ker ( φ ) . iitrpr.pdf