abstractlet w be a non-empty subset of a free group. the automorphism of a group g is said to be a marginal automorphism, if for all x in g,x^−1alpha(x) in w^*(g), where w^*(g) is the marginal subgroup of g.in this paper, we give necessary and sufficient condition for a purelynon-abelian p-group g, such that the set of all marginal automorphismsof g forms an elementary abelian p-group.

AbstractLet W be a non-empty subset of a free group. The automorphism of a group G is said to be a marginal automorphism, if for all x in G,x^−1alpha(x) in W^*(G), where W^*(G) is the marginal subgroup of G.In this paper, we give necessary and sufficient condition for a purelynon-abelian p-group G, such that the set of all marginal automorphismsof G forms an elementary abelian p-group.

Let A and G be finite groups of relatively prime orders and assume that A acts on G via automorphisms. We study how certain conditions on G imply its solvability when we assume the existence of a unique A-invariant Sylow p-subgroup for p equal to 2 or 3. Mathematics Subject Classification (2010). Primary 20D20; Secondary 20D45.

In this article, a Blackburn group refers to a finite non-Dedekind group for which the intersection of all nonnormal subgroups is not the trivial subgroup. By completing the arguments of M. Hertweck, we show that all conjugacy class preserving automorphisms of Blackburn groups are inner automorphisms. 2000 Mathematics subject classification: primary 20D45; secondary 16S34.

Let S be a closed oriented surface S of genus g ≥ 0 with m ≥ 0 marked points (punctures) and 3g − 3 + m ≥ 2. This is a survey of recent results on actions of the mapping class group of S which led to a geometric understanding of this group. Mathematics Subject Classification (2010). Primary 30F60, Secondary 20F28, 20F65, 20F69

The description of the automorphism group of group 〈a, b; [am, bn] = 1〉 (m, n > 1) in terms of generators and defining relations is given. This result is applied to prove that any normal automorphism of every such group is inner. 2000 Mathematics Subject Classification: primary 20F28, 20F05; secondary 20E06.

Duality and chirality are examples of operations of order 2 on hypermaps. James showed that the groups of all operations on hypermaps and on oriented hypermaps can be identified with the outer automorphism groups Out∆ ∼= PGL2(Z) and Out∆ ∼= GL2(Z) of the groups ∆ = C2 ∗ C2 ∗ C2 and ∆ + = F2. We will consider the elements of finite order in these two groups, and the operations they induce. MSC c...

In this paper we first find the automorphism group of the direct product of n copies of an indecomposable non-abelian group. We describe the automorphism group as matrices with entries which are homomorphisms between the n direct factors. We then use this description with a generalization of a result by Bidwell, Curran, and McCaughan on Aut (H×K), where H and K have no common direct factor, to ...

The c-dimension of a group is the maximum length of a chain of nested centralizers. It is proved that a periodic locally soluble group of finite cdimension k is soluble of derived length bounded in terms of k, and the rank of its quotient by the Hirsch–Plotkin radical is bounded in terms of k. Corollary: a pseudo-(finite soluble) group of finite c-dimension k is soluble of derived length bounde...

We prove that the Bianchi groups, that is the groups PSL(2,O) where O is the ring of integers in an imaginary quadratic number field, are good. This is a property introduced by J.P. Serre which relates the cohomology groups of a group to those of its profinite completion. We also develop properties of goodness to be able to show that certain natural central extensions of Fuchsian groups are res...