Class-preserving Automorphisms Research Articles

Tate–Shafarevich groups and algebras

The Tate–Shafarevich set of a group [Formula: see text] defined by Takashi Ono coincides, in the case where [Formula: see text] is finite, with the group of outer class-preserving automorphisms of [Formula: see text] introduced by Burnside. We consider analogs of this important group-theoretic object for Lie algebras and associative algebras and establish some new structure properties thereof. We also discuss open problems and eventual generalizations to other algebraic structures.

On the autoconjugate probability of finite groups

Let G be a finite group and let denote the probability that two randomly selected elements of G are conjugate. Two elements of G are called autoconjugate if one is the image of other under an automorphism of G. Let denote the probability that two randomly selected elements of G are autoconjugate. We calculate for some finite groups and some lower bounds of . We also prove that if and only if , where denotes the group of all class-preserving automorphisms of G. This gives another perspective on a question of Avinoam Mann.

Class-preserving automorphisms of certain HNN extensions

We consider HNN extensions [Formula: see text], where [Formula: see text] are normal subgroups of [Formula: see text] and [Formula: see text]. We show that class-preserving automorphisms of those HNN extensions are all inner if [Formula: see text] satisfies certain conditions. From Grossman’s result, it follows that outer automorphism groups of such conjugacy separable HNN extensions are residually finite.

The Normalizer Property for Finite Groups Whose Sylow 2-Subgroups are Abelian

In this paper we mainly investigate the Coleman automorphisms and class-preserving automorphisms of finite AZ-groups and finite groups related to AZ-groups. For example, we first prove that $$Out_c(G)$$ of an AZ-group G must be a $$2'$$ -group and therefore the normalizer property holds for G. Then we find some classes of finite groups such that the intersection of their outer class-preserving automorphism groups and outer Coleman automorphism groups is $$2'$$ -groups, and therefore, the normalizer property holds for these kinds of finite groups. Finally, we show that the normalizer property holds for the wreath products of AZ-groups by rational permutation groups under some conditions.

On the Probability That an Automorphism Fixes a Group Element

Let G be a finite group. Denote by P(G) the probability that two elements of G, selected randomly and with replacement, commute, and by PA(G) the probability that a randomly chosen automorphism of G fixes a randomly chosen element of G. It is known that for any nonabelian group. We show that for any noncyclic group, and then prove that this bound is not sharp. We also prove that if and only if , where denotes the group of all conjugacy class-preserving automorphisms of G. This gives another perspective on a question of Avinoam Mann.

On Bogomolov multiplier and Ш-rigidity of groups

We show the existence of groups with trivial Bogomolov multiplier such that the Bogomolov multiplier of a central product of these groups is non-trivial. We then prove that finite quasisimple groups and freest special p-groups are Ш-rigid in the sense that all their class-preserving automorphisms are inner. As an application we show the existence of non-nilpotent Ш-rigid groups and Ш-rigid p-groups of odd order having non-trivial Bogomolov multiplier.

The group of automorphisms of an elementary-abelian-over-cyclic regular wreath product p-group

Let W denote the regular wreath product finite group where C is a cyclic p-group and is an elementary abelian p-group. Let A denote the subgroup of consisting of those automorphisms that act trivially on , where B is the base group. We determine A by describing where each of its elements map a certain generating set for W. We find that A is as large as possible in a certain sense. We determine some information about the subgroup structure of A, and we prove that every class-preserving automorphism of W is an inner automorphism of W.

Class-preserving coleman automorphisms of permutational wreath products with 2-closed base groups

Let [Formula: see text] be a nontrivial [Formula: see text]-closed group and let [Formula: see text] be an arbitrary permutation group on a finite set [Formula: see text]. Let [Formula: see text] be the corresponding permutational wreath product of [Formula: see text] by [Formula: see text]. It is shown that every class-preserving Coleman automorphism of [Formula: see text]-power order of [Formula: see text] is inner. As a direct consequence, it is obtained that the normalizer property holds for [Formula: see text]. Further, it is shown that every class-preserving Coleman automorphism of [Formula: see text] is inner whenever [Formula: see text] is nilpotent. Our results generalize some known ones.

On nilpotency of the group of outer class-preserving automorphisms of a group

Let G be a group and Zj(G), for j ≥ 0, be the jth term in the upper central series of G. We prove that if Outc(G/Zj(G)), the group of outer class-preserving automorphisms of G/Zj(G), is nilpotent of class k, then Outc(G) is nilpotent of class at most j + k. Moreover, if Outc(G/Zj(G)) is a trivial group, then Outc(G) is nilpotent of class at most j. As an application we prove that if γi(G)/γi(G) ∩ Zj(G) is cyclic then Outc(G) is nilpotent of class at most i + j, where γi(G), for i ≥ 1, denotes the ith term in the lower central series of G. This extends an earlier work of the author, where this assertion was proved for j = 0. We also improve bound on the nilpotency class of Outc(G) for some classes of nilpotent groups G.

Class-preserving automorphisms of finite p-groups II

Let G be a finite group minimally generated by d(G) elements and Aut c (G) denote the group of all (conjugacy) class-preserving automorphisms of G. Continuing our work [Class preserving automorphisms of finite p-groups, J. London Math. Soc. 75 (2007), 755–772], we study finite p-groups G such that |Aut c (G)| = |γ2(G)| d(G), where γ2(G) denotes the commutator subgroup of G. If G is such a p-group of class 2, then we show that d(G) is even, 2d(γ2(G)) ≤ d(G) and G/ Z(G) is homocyclic. When the nilpotency class of G is larger than 2, we obtain the following (surprising) results: (i) d(G) = 2. (ii) If |γ2(G)/γ 3(G)| > 2, then |Aut c (G)| = |γ2(G)| d(G) if and only if G is a 2-generator group with cyclic commutator subgroup, where γ 3(G) denotes the third term in the lower central series of G. (iii) If |γ2(G)/γ 3(G)| = 2, then |Aut c (G)| = |γ2(G)| d(G) if and only if G is a 2-generator 2-group of nilpotency class 3 with elementary abelian commutator subgroup of order at most 8. As an application, we classify finite nilpotent groups G such that the central quotient G/ Z(G) of G by its center Z(G) is of the largest possible order. For proving these results, we introduce a generalization of Camina groups and obtain some interesting results. We use Lie theoretic techniques and computer algebra system ‘Magma’ as tools.

Class-preserving automorphisms of some finite p-groups

Let G be a finite p-group of order p5, where p is a prime. We give necessary and sufficient conditions on G such that G has a non-inner class-preserving automorphism. As a consequence, we give short and alternate proofs of results of Yadav (§5 of Proc. Indian Acad. Sci. (Math. Sci.)118 (2008) 1–11) and Kalra and Gumber (Theorem 4.2 of Indian J. Pure Appl. Math.44 (2013) 711–725).

Class-preserving automorphisms of certain HNN extensions

We show that, for any integers γ,β, class-preserving automorphisms of the HNN extension 〈B,t:t−1hγt=hβ〉 of a finitely generated nilpotent or free group B are all inner. It follows from Grossman's result that outer automorphism groups of such conjugacy separable HNN extensions are residually finite.

Bogomolov multiplier, double class-preserving automorphisms, and modular invariants for orbifolds

We describe the group $Aut_{br}^1({\cal Z}(G))$Autbr1(Z(G)) of braided tensor autoequivalences of the Drinfeld centre of a finite group G isomorphic to the identity functor (just as a functor). We prove that the semi-direct product Out2 − cl(G)⋉B(G) of the group of double class preserving automorphisms and the Bogomolov multiplier of G is a subgroup of $Aut_{br}^1({\cal Z}(G))$Autbr1(Z(G)). An automorphism of G is double class preserving if it preserves conjugacy classes of pairs of commuting elements in G. The Bogomolov multiplier B(G) is the subgroup of its Schur multiplier H2(G, k*) of classes vanishing on abelian subgroups of G. We show that elements of $Aut^1_{br}({\cal Z}(G))$Autbr1(Z(G)) give rise to different realisations of the charge conjugation modular invariant for G-orbifolds of holomorphic conformal field theories.

The Bogomolov multiplier of rigid finite groups

The Bogomolov multiplier of a finite group G is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of G. This invariant of G plays an important role in birational geometry of quotient spaces V/G. We show that in many cases the vanishing of the Bogomolov multiplier is guaranteed by the rigidity of G in the sense that it has no outer class-preserving automorphisms.

CLASS-PRESERVING AUTOMORPHISMS OF GENERALIZED FREE PRODUCTS AMALGAMATING A CYCLIC NORMAL SUBGROUP

In general, a class-preserving automorphism of generalized free products of nilpotent groups, amalgamating a cyclic normal subgroup of order 8, need not be an inner automorphism. We prove that every class-preserving automorphism of generalized free products of nitely generated nilpotent groups, amalgamating a cyclic normal subgroup of order less than 8, is inner.

Class-preserving automorphisms and inner automorphisms of certain tree products of groups

In general, a class-preserving automorphism of generalized free products of nilpotent groups need not be an inner automorphism. We prove that every class-preserving automorphism of tree products of finitely generated nilpotent or free groups, amalgamating infinite cyclic subgroups, is inner. It follows that the outer automorphism groups of such tree products, amalgamating infinite cyclic subgroups, are residually finite.

Class-preserving automorphisms and the normalizer property for Blackburn groups

For a group G, let U be the group of units of the integral group ring ZG. The group G is said to have the normalizer property if NUðG Þ¼ ZðUÞG. It is shown that Blackburn groups have the normalizer property. These are the groups which have non-normal finite subgroups, with the intersection of all of them being non-trivial. Groups G for which class- preserving automorphisms are inner automorphisms, OutcðG Þ¼ 1, have the normalizer prop- erty. Recently, Herman and Li have shown that OutcðG Þ¼ 1 for a finite Blackburn group G. We show that OutcðG Þ¼ 1 for the members G of certain classes of metabelian groups, from which the Herman-Li result follows. Together with recent work of Hertweck, Iwaki, Jespers and Juriaans, our main result implies that, for an arbitrary group G, the group ZyðUÞ of hypercentral units of U is contained in ZðUÞG.

Class-Preserving Coleman Automorphisms of Finite Groups

Let G be a finite group whose Sylow 2-subgroups are either cyclic, dihedral, or generalized quaternion. It is shown that a class-preserving automorphism of G of order a power of 2 whose restriction to any Sylow subgroup of G equals the restriction of some inner automorphism of G is necessarily an inner automorphism. Interest in such automorphisms arose from the study of the isomorphism problem for integral group rings, see [6, 7, 13, 14].

Class-Preserving Automorphisms of Finite Groups

We construct a family F of Frobenius groups having abelian Sylow subgroups and non-inner, class-preserving automorphisms. We show that any A-group (that is, a finite solvable group with abelian Sylow subgroups) has a sub-quotient belonging to F provided it has a non-inner, class-preserving automorphism. As a consequence, we obtain that for metabelian A-groups, or A-groups with elementary abelian Sylow subgroups, class-preserving automorphisms are necessarily inner automorphisms. The same is true for a finite group whose Sylow subgroups of odd order are all cyclic, and whose Sylow 2-subgroups are either cyclic, dihedral, or generalized quaternion. Some applications are given.