Infinite groups with a finiteness conditions on non-abelian - PowerPoint PPT Presentation

Background - L ab ( G ) , L non − ab ( G ) Example If P = ab is the property to be abelian , then L ab ( G ) is finite ⇔ G is finite. Remark Tarski monsters and more generally minimal non-abelian groups are groups G with L non − ab ( G ) finite . Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

Background - L ab ( G ) Let P = ab be the property to be abelian . Groups G in which L ab ( G ) , ordered by inclusion, has M ax or M in have been firstly studied respectively by A.I. Mal’cev in 1956 and O.J. Schmidt in 1945. A.I. Mal’cev, On certain classes of infinite soluble groups, Mat. Sb. 28 (1951), 567-588 (Russian), Amer. Math. Soc. Transl. (2) 2 (1956), 1-21. O.J. Schmidt, Infinite soluble groups, Mat. Sb. , 17(59) (1945), 145-162 (Russian). Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

Background - L non − ab ( G ) Groups G in which L non − ab ( G ) has M ax have been studied by L.A. Kurdachenko and D.I. Zaicev in 1991. L.A. Kurdachenko, D.I. Zaicev, Groups with the maximum condition for non-abelian subgroups, Ukrain. Mat. Zh. 43 (1991), 925-930 (Russian), English transl., Ukrainian Math. J. 43 (1991), 863-868. Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

Background - L non − ab ( G ) Groups in which L non − ab ( G ) has M in have been studied by S.N. Černikov in 1964 and 1967. S.N. Černikov, Infinite groups with prescribed properties of their systems of infinite subgroups, Dokl. Akad. Nauk SSSR , 159 (1964) 759-760 (Russian), Soviet Math. Dokl. , 5 (1964) 1610-1611. S.N. Černikov, Groups with given properties of systems of infinite subgroups, Ukrain. Mat. Ž , 19 (1967) 111-131 (Russian), Ukrainian Math. J. , 19 (1967) 715-731. Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

Background - L N ( G ) Let N be the property to be nilpotent . Groups G in which L N ( G ) has M ax have been studied by M.R. Dixon and L.A. Kurdachenko in 2001. M.R. Dixon, L.A. Kurdachenko, Groups with the maximum condition on non-nilpotent subgroups, J. Group Theory 4 (2001), 75-87. M.R. Dixon, L.A. Kurdachenko, Locally nilpotent groups with the maximum condition on non-nilpotent subgroups, Glasgow Math. J. 43 (2001), 85-102. Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

Paper L.A. Kurdachenko, P. L., M. Maj, I.Ya Subbotin Groups with finitely many types of non-isomorphic non-abelian subgroups submitted . Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

A new problem We study a quite different finiteness condition on L P ( G ) and L non −P ( G ) . Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

Definition Let G be a group and let M be a family of subgroups of G . Definition Consider the equivalence relation in M given by H ≃ K , with H , K ∈ M . Call the isomorphic type Itype M of M any set of representatives of all equivalence classes in M . Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

Problem Let G be a group and let M be a family of subgroups of G . Definitions Consider the equivalence relation in M given by H ≃ K , with H , K ∈ M . Call the isomorphic type Itype M of M any set of representatives of all equivalence classes in M . We study groups G in which Itype M is finite . Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

First remarks - | Itype L ( G ) | Let G be a group. Remark If G is non-trivial, then G , { 1 } ∈ Itype L ( G ) . Thus | Itype L ( G ) | � 2 . Proposition | Itype L ( G ) | = 2 ⇔ either | G | a prime or G infinite cyclic. Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

First remarks - | Itype L ( G ) | Proposition | Itype L ( G ) | = 2 ⇔ either | G | a prime or G infinite cyclic. Proof. If G is cyclic, either infinite or of prime order, then obviously Itype L ( G ) = {{ 1 } , G } . Conversely, assume | Itype L ( G ) | = 2, so Itype L ( G ) = {{ 1 } , G } . Then, for any x ∈ G − { 1 } , we have < x > ≃ G . Therefore G is cyclic. If G is finite, then there is an element y ∈ G of order a prime and G ≃ < y > , as required. // Problem What about Itype L ab ( G ) ? Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

First remarks - | Itype L ab ( G ) | Problem What about | Itype L ab ( G ) | ? Remark Obviously, if G � = { 1 } , then { 1 } , < x > ∈ Itype L ab ( G ) , where x ∈ G − { 1 } . Therefore | Itype L ab ( G ) | � 2. But Tarski monsters T have | Itype L ab ( T ) | = 2 . Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

First remarks - | Itype L ( G ) | , | Itype L ab ( G ) | Remark Using a result due to V.S. Charin it follows that if a group G is such that Itype L ( G ) or Itype L ab ( G ) is finite , then every abelian subgroup of G is minimax. Definition A group G is said to be minimax if it has a finite series whose factors satisfy M in or M ax . V.S. Charin, On soluble groups of type A 4 , Mat. Sbornik 52 (1960), no. 3, 895-914. Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

First remarks - | Itype L non − ab ( G ) | Problem What about Itype L non − ab ( G ) ? Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

First remarks - | Itype L non − ab ( G ) | Problem What about | Itype L non − ab ( G ) | ? Groups G with | Itype L non − ab ( G ) | = 1 have been studied by H. Smith and J. Wiegold in 1997. H. Smith, J. Wiegold, Groups which are isomorphic to their non-abelian subgroups , Rend. Math. Univ. Padova 97 (1997), 7-16. Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

First remarks - | Itype L non − ab ( G ) | Groups G with | Itype L non − ab ( G ) | = 1 have been studied by H. Smith and J. Wiegold in 1997. Among other results they proved: Theorem Let G be a soluble group. If G is isomorphic to every non abelian subgroup, then G contains an abelian normal subgroup of prime index. Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

New results - Itype L non − ab ( G ) finite Jointly with L.A. Kurdachenko , M. Maj and I.Ya Subbotin we studied groups G such that Itype L non − ab ( G ) is finite . Remark Because of Tarski monsters, we have to assume something about G . Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

Examples - Itype L non − ab ( G ) finite Example Let D ∞ be the infinite dihedral group: D ∞ = < a , b | b 2 = 1 , a b = a − 1 > . Then | Itype L non − ab ( D ∞ ) | = 1. More generally: Remark If G is a finitely generated abelian-by-finite group, then Itype L non − ab ( G ) is finite . Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

Examples - Itype L non − ab ( G ) infinite Example Let A = Dr p ∈P A p , where A p � = { 1 } is an abelian p -group, and let G = A ⋊ < b > , where b 2 = 1 , a b = a − 1 , for any a ∈ A . Then Itype L non − ab ( G ) is infinite. For, A p < b > is non-abelian for any p ∈ P and A p < b > �≃ A q < b > for any p � = q . Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

New results - Itype L non − ab ( G ) finite Lemma 1 Let G be a group with Itype L non − ab ( G ) finite . If K is an infinite locally finite subgroup of G , then K is abelian . Proof. Suppose that K is non-abelian. Being locally finite, K includes a finite non-abelian subgroup F . Then G has an ascending chain F = F 0 ≤ F 1 ≤ · · · ≤ F n ≤ F n + 1 ≤ . . . of finite subgroups such that | F n | < | F n + 1 | for each n ∈ N . But in this case, the subgroups F n and F m cannot be isomorphic for n , m ∈ N , n � = m , and we obtain a contradiction. // Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

New results - Itype L non − ab ( G ) finite Lemma 2 Let G be a group with Itype L non − ab ( G ) finite . If G is locally nilpotent, then G is nilpotent . Proof. W.l.o.g. we can assume that G is non-abelian. Then G includes a non-abelian finitely generated subgroup K . Suppose that G is not nilpotent. Then G has an ascending chain K = K 0 ≤ K 1 ≤ · · · ≤ K n ≤ K n + 1 ≤ · · · of finitely generated subgroups such that ncl ( K n ) < ncl ( K n + 1 ) for each n ∈ N . But in this case the subgroups K n and K m cannot be isomorphic for n , m ∈ N , n � = m . This contradiction shows that G must be nilpotent. // Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

New results - Itype L non − ab ( G ) finite Proposition 1 Let G be a group in which L non − ab ( G ) is finite . Let A be an infinite abelian periodic subgroup of G . Then N G ( A ) = C G ( A ) . Sketch of the proof. Let x be an arbitrary element of N G ( A ) . We have A = Dr p ∈ Π( A ) A p , where A p is a Sylow p -subgroup of A , p ∈ Π( A ) . Clearly every subgroup A p is < x > -invariant. If A p is infinite, then x ∈ C G ( A p ) . In particular, A p ≤ FC ( < A , x > ) . If A p is finite, then again we have the inclusion A p ≤ FC ( < A , x > ) . It follows that A ≤ FC ( < A , x > ) . In this case x ∈ C G ( A ) . // Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

New results - Itype L non − ab ( G ) finite Proposition 2 Let G be a group, in which L non − ab ( G ) is finite . Let A be an abelian torsion-free subgroup of G . If N G ( A ) � = C G ( A ) , then A is minimax. Sketch of the proof. Since N G ( A ) � = C G ( A ) , we can choose an element x ∈ N G ( A ) \ C G ( A ) . It is possible to see that A has finite 0-rank. Let { a 1 , · · · , a n } be a maximal Z -independent subset of A . Then the subgroup A j = < a j > < x > is minimax. It follows that x k j ∈ C G ( A j ) , 1 ≤ j ≤ n . Put k = k 1 · · · k n , then x k ∈ C G ( < a 1 , · · · , a n > ). Since A is torsion-free and A is the pure envelope of < a 1 , · · · , a n > , we have C < x > ( < a 1 , · · · , a n > ) = C < x > ( A ) . Then we obtain that A is minimax. // Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

New results - Itype L non − ab ( G ) finite Definition A group G is called radical if there exists an ascending series of G with locally nilpotent factors. Examples Any locally nilpotent group is a radical group. Any soluble group is a radical group. Definition A group G is called generalized radical if G has an ascending series whose factors are either locally nilpotent or locally finite. Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

New results - Itype L non − ab ( G ) finite Theorem A Let G be a non-abelian locally generalized radical group. If Itype L non − ab ( G ) is finite , then G is a minimax, abelian-by-finite group, with Tor ( G ) finite. Definition Tor ( G ) is the maximal normal torsion subgroup of G . Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

New results - Itype L non − ab ( G ) finite Definition A group G is called generalized coradical if there exists a descending series of G whose factors are either locally nilpotent or locally finite. Theorem B Let G be a non-abelian generalized coradical group. If Itype L non − ab ( G ) is finite , then G is a minimax, abelian-by-finite group with Tor ( G ) finite. Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...

Infinite groups with a finiteness conditions on non-abelian - PowerPoint PPT Presentation

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