Structural criteria in factorised groups via conjugacy class sizes - PowerPoint PPT Presentation

Structural criteria in factorised groups via conjugacy class sizes V´ ıctor Manuel Ortiz-Sotomayor Universitat Polit` ecnica de Val` encia, Spain G ROUPS S T A NDREWS 2017 Birmingham, August 2017 Joint work with: ⊲ Mar´ ıa Jos´ e Felipe ⊲ Ana Mart´ ınez-Pastor

I NTRODUCTION

Introduction Square-free class sizes Prime power class sizes Preliminaries All groups considered will be finite ⊲ Factorised groups: G = AB , where A and B are subgroups of a group G

Introduction Square-free class sizes Prime power class sizes Preliminaries All groups considered will be finite ⊲ Factorised groups: G = AB , where A and B are subgroups of a group G Questions. 1. How the structure of the factors A and B affects the structure of the whole group G ? 2. How the structure of the group G affects the structure of A and B ?

Introduction Square-free class sizes Prime power class sizes Preliminaries All groups considered will be finite ⊲ Factorised groups: G = AB , where A and B are subgroups of a group G Questions. 1. How the structure of the factors A and B affects the structure of the whole group G ? 2. How the structure of the group G affects the structure of A and B ? Theorem (Fitting, 1938). If G = AB is the product of the nilpotent normal subgroups A and B , then G is nilpotent.

Introduction Square-free class sizes Prime power class sizes Preliminaries All groups considered will be finite ⊲ Factorised groups: G = AB , where A and B are subgroups of a group G Questions. 1. How the structure of the factors A and B affects the structure of the whole group G ? 2. How the structure of the group G affects the structure of A and B ? Theorem (Fitting, 1938). If G = AB is the product of the nilpotent normal subgroups A and B , then G is nilpotent. Theorem (Kegel–Wielandt, 1962, 1958). If G = AB is the product of the nilpotent subgroups A and B , then G is soluble.

Introduction Square-free class sizes Prime power class sizes Preliminaries we denote x G := { x g : g ∈ G } , and its size is � � x G � ⊲ Conjugacy class of x ∈ G : �

Introduction Square-free class sizes Prime power class sizes Preliminaries we denote x G := { x g : g ∈ G } , and its size is � � x G � ⊲ Conjugacy class of x ∈ G : � New topic ⇓ � where x ∈ A ∪ B . � � x G � G = AB and to impose arithmetical conditions on

Introduction Square-free class sizes Prime power class sizes Preliminaries we denote x G := { x g : g ∈ G } , and its size is � � x G � ⊲ Conjugacy class of x ∈ G : � New topic ⇓ � where x ∈ A ∪ B . � � x G � G = AB and to impose arithmetical conditions on Arithmetical conditions. 1. Square-free class sizes. 2. Prime power class sizes.

Introduction Square-free class sizes Prime power class sizes Preliminaries we denote x G := { x g : g ∈ G } , and its size is � � x G � ⊲ Conjugacy class of x ∈ G : � New topic ⇓ � where x ∈ A ∪ B . � x G � � G = AB and to impose arithmetical conditions on Arithmetical conditions. 1. Square-free class sizes. 2. Prime power class sizes. Some notes. � does not divide � � x A � � � x G � 1. In general, if x ∈ A , then � .

Introduction Square-free class sizes Prime power class sizes Preliminaries we denote x G := { x g : g ∈ G } , and its size is � � x G � ⊲ Conjugacy class of x ∈ G : � New topic ⇓ � where x ∈ A ∪ B . � x G � � G = AB and to impose arithmetical conditions on Arithmetical conditions. 1. Square-free class sizes. 2. Prime power class sizes. Some notes. � does not divide � � x A � � � x G � 1. In general, if x ∈ A , then � . 2. If N � G , then generally it is not prefactorised , that is, N � = ( N ∩ A )( N ∩ B ) .

Introduction Square-free class sizes Prime power class sizes Preliminaries we denote x G := { x g : g ∈ G } , and its size is � � x G � ⊲ Conjugacy class of x ∈ G : � New topic ⇓ � where x ∈ A ∪ B . � � x G � G = AB and to impose arithmetical conditions on Arithmetical conditions. 1. Square-free class sizes. 2. Prime power class sizes. Some notes. � does not divide � � x A � � x G � � 1. In general, if x ∈ A , then � . 2. If N � G , then generally it is not prefactorised , that is, N � = ( N ∩ A )( N ∩ B ) . 3. Loss of information about class sizes of “diagonal” elements: If g ∈ G = AB , then g G = ( ab ) G � = a G b G , and � � g G � � is not related with � � a G � � and � � b G � � .

Introduction Square-free class sizes Prime power class sizes Starting point Theorem. Let G = AB , where A and B are permutable subgroups of G . � is square-free for every x ∈ A ∪ B . � � x G � Suppose that Then G is supersoluble. Liu, X., Wang, Y., and Wei, H., Notes on the length of conjugacy classes of finite groups , J. Pure and Applied Algebra, 196: 111-117, 2005.

Introduction Square-free class sizes Prime power class sizes Starting point Theorem. Let G = AB , where A and B are permutable subgroups of G . � is square-free for every x ∈ A ∪ B . � � x G � Suppose that Then G is supersoluble. Liu, X., Wang, Y., and Wei, H., Notes on the length of conjugacy classes of finite groups , J. Pure and Applied Algebra, 196: 111-117, 2005. Remark. � divides � � a A � � � a G � The factors A and B inherit the class size hypotheses, i.e., � , ∀ a ∈ A .

Introduction Square-free class sizes Prime power class sizes Starting point Theorem. Let G = AB , where A and B are permutable subgroups of G . � is square-free for every x ∈ A ∪ B . � � x G � Suppose that Then G is supersoluble. Liu, X., Wang, Y., and Wei, H., Notes on the length of conjugacy classes of finite groups , J. Pure and Applied Algebra, 196: 111-117, 2005. Remark. � divides � � a A � � � a G � The factors A and B inherit the class size hypotheses, i.e., � , ∀ a ∈ A . Question. To what extent can the permutability hypothesis on the factors be weakened?

Introduction Square-free class sizes Prime power class sizes Starting point Theorem. Let G = AB , where A and B are permutable subgroups of G . � is square-free for every x ∈ A ∪ B . � � x G � Suppose that Then G is supersoluble. Liu, X., Wang, Y., and Wei, H., Notes on the length of conjugacy classes of finite groups , J. Pure and Applied Algebra, 196: 111-117, 2005. Corollary. � is square-free for every x ∈ G . � � x G � Suppose that Then G is supersoluble (and both G / F ( G ) and G ′ are cyclic with square-free orders). Chillag, D., and Herzog, M., On the length of the conjugacy classes of finite groups , J. Algebra, 131: 110-125, 1990. Cossey, J., and Wang, Y., Remarks on the length of conjugacy classes of finite groups , Comm. Algebra, 27: 4347-4353, 1999.

Introduction Square-free class sizes Prime power class sizes Starting point Theorem. Let G = AB , where A and B are permutable subgroups of G . � is square-free for every x ∈ A ∪ B . � � x G � Suppose that Then G is supersoluble. Liu, X., Wang, Y., and Wei, H., Notes on the length of conjugacy classes of finite groups , J. Pure and Applied Algebra, 196: 111-117, 2005. Corollary. � is square-free for every x ∈ G . � � x G � Suppose that Then G is supersoluble (and both G / F ( G ) and G ′ are cyclic with square-free orders). Chillag, D., and Herzog, M., On the length of the conjugacy classes of finite groups , J. Algebra, 131: 110-125, 1990. Cossey, J., and Wang, Y., Remarks on the length of conjugacy classes of finite groups , Comm. Algebra, 27: 4347-4353, 1999. Question. Can we obtain further information for factorised groups?

Introduction Square-free class sizes Prime power class sizes Permutability properties G = AB : A , B supersoluble + A , B � G � G supersoluble Question. G = AB : A , B supersoluble + permutability properties + additional conditions. Then G is supersoluble.

Introduction Square-free class sizes Prime power class sizes Permutability properties G = AB : A , B supersoluble + A , B � G � G supersoluble Question. G = AB : A , B supersoluble + permutability properties + additional conditions. Then G is supersoluble. Theorem. Let G = AB with A and B supersoluble, and assume that G ′ is nilpotent. If A and B are mutually permutable, then G is supersoluble. Asaad, M., and Shaalan, A., On the supersolvability of finite groups , Arch. Math. (Basel), 53: 318-326, 1989. Definition. A and B are mutually permutable if A permutes with every subgroup of B and B permutes with every subgroup of A .

S QUARE - FREE CLASS SIZES

Introduction Square-free class sizes Prime power class sizes Some results Theorem. Let G = AB such that A and B are mutually permutable. � is square-free for every element x ∈ A ∪ B . Then we have: � x G � � Assume ⊲ G is supersoluble. ⊲ G ′ is abelian with elementary abelian Sylow subgroups. ⊲ G / F ( G ) has elementary abelian Sylow subgroups. ⊲ The structure of the Sylow p -subgroups of G / F ( G ) is either C p or C p × C p , ∀ p . Felipe, M. J., Mart´ ınez-Pastor, A., and Ortiz-Sotomayor, V. M., Square-free class sizes in products of groups , submitted.