On central automorphisms that fix the centre elementwise

Let G be a group and let Aut _c (G) be the group of central automorphisms of G. Let be the set of all central automorphisms of G fixing Z(G) elementwise. In this paper we prove that if G is a finite p-group, then = Inn(G) if and only if G is abelian or G is nilpotent of class 2 and Z(G) is cyclic. This work was supported in part by the Center of Excellence for Mathematics, University of Isfahan, Iran. Received: 30 October 2006     

Transfer and Tate’s theorem

Let H be a subgroup of a finite group G, and assume that p is a prime that does not divide |G : H|. In favorable circumstances, one can use transfer theory to deduce that the largest abelian p-groups that occur as factor groups of G and of H are isomorphic. When this happens, Tate’s theorem guarantees that the largest not-necessarily-abelian p-groups that occur as factor groups of G and H are isomorphic. Known proofs of Tate’s theorem involve cohomology or character theory, but in this paper, a new elementary proof is given. It is also shown that the largest abelian p-factor group of G is always isomorphic to a direct factor of the largest abelian p-factor group of H. Received: 17 June 2008     

On δ-permutability of maximal subgroups of Sylow subgroups of finite groups

In this paper, we get the main theorem: Let p be a prime dividing the order of G and , where and H is p ^′-Hall subgroup of G. Let δ be a complete set of Sylow subgroups of H. If G satisfies the following conditions: i) is a p-group; ii) for any maximal M of P, M is δ-permutable in H, then G is p-nilpotent. Some known results are generalized. Received: 12 September 2007, Revised: 28 February 2008     

On automorphisms of A-groups

Let G be an A-group (i.e. a group in which xx ^α  = x ^α x for all and let denote the subgroup of Aut(G) consisting of all automorphisms that leave invariant the centralizer of each element of G. The quotient is an elementary abelian 2-group and natural analogies exist to suggest that it might always be trivial. It is shown that, in fact, for any odd prime p and any positive integer r, there exist infinitely many finite pA-groups G for which has rank r. Received: 23 March 2008, Revised: 20 May 2008     

The Schur property for subgroup lattices of groups

A classical theorem of Schur states that if the centre of a group G has finite index, then the commutator subgroup G′ of G is finite. A lattice analogue of this result is proved in this paper: if a group G contains a modularly embedded subgroup of finite index, then there exists a finite normal subgroup N of G such that G/N has modular subgroup lattice. Here a subgroup M of a group G is said to be modularly embedded in G if the lattice is modular for each element x of G. Some consequences of this theorem are also obtained; in particular, the behaviour of groups covered by finitely many subgroups with modular subgroup lattice is described. Received: 16 October 2007, Final version received: 22 February 2008     

The Kernel of the Adjoint Representation of a p-Adic Lie Group Need Not Have an Abelian Open Normal Subgroup

Let G be a p-adic Lie group with Lie algebra 𝔤 and Ad: G → Aut(𝔤) be the adjoint representation. It was claimed in the literature that the kernel K?ker(Ad) always has an abelian open normal subgroup. We show by means of a counterexample that this assertion is false. It can even happen that K = G, but G has no abelian subnormal subgroup except for the trivial group. The arguments are based on auxiliary results on subgroups of free products with central amalgamation.     

The influence of $\pi$-quasinormality of some subgroups of a finite group

A subgroup H of G is said to be $\pi$-quasinormal in G if it permute with every Sylow subgroup of G. In this paper, we extend the study on the structure of a finite group under the assumption that some subgroups of G are $\pi$-quasinormal in G. The main result we proved in this paper is the following:Theorem 3.4. Let ${\cal F}$ be a saturated formation containing the supersolvable groups. Suppose that G is a group with a normal subgroup H such that $G/H \in {\cal F}$, and all maximal subgroups of any Sylow subgroup of $F^{*}(H)$ are $\pi$-quasinormal in G, then $G \in {\cal F}$. Received: 10 May 2002     

Recurrent random walks on homogeneous spaces of p-adic algebraic groups of polynomial growth

Let G be a p-adic algebraic group of polynomial growth and H be a closed subgroup of G. We prove the growth conjecture for the homogeneous space G/H, that is, G/H supports a recurrent random walk if and only if G/H has polynomial growth of degree atmost two. Received: 23 November 2007     

Automorphisms of direct products of finite groups

This paper shows that if H and K are finite groups with no common direct factor and G = H × K, then the structure and order of Aut G can be simply expressed in terms of Aut H, Aut K and the central homomorphism groups Hom (H, Z(K)) and Hom (K, Z(H)). Received: 18 April 2005; revised: 9 June 2005     

Automorphisms of blocks of group algebras and character values

Let G be a finite group, χ an irreducible complex character of G and A(χ) the block ideal of the group algebra ℚG relatedℴ χ. The aim of this paper is to study the group Aut (A(χ)) of all ring (or ℚ-algebra) automorphisms of A(χ). Especially we are interested in the existence of subgroups of Aut (A(χ)), which are isomorphic to a given subgroup Γ of the Galois group of the field of character values ℚ(χ) over the rationals. In this context we prove some results related to character values.     

A Family of Nonnormal Cayley Digraphs

We call a Cayley digraph Γ = Cay(G, S) normal for G if G _R , the right regular representation of G, is a normal subgroup of the full automorphism group Aut(Γ) of Γ. In this paper we determine the normality of Cayley digraphs of valency 2 on nonabelian groups of order 2p ² (p odd prime). As a result, a family of nonnormal Cayley digraphs is found. Received February 23, 1998, Revised September 25, 1998, Accepted October 27, 1998     

On the Converse of the Theory of Groups Acting on Trees with Inversions

In this paper we show that if G is a group acting on a graph X with inversions such that G has a presentation induced by a fundamental domain for the action of G on X, then X is a tree. Received: January 3, 2007., Revised: August 10, 2007 and May 3, 2008., Accepted: October 17, 2008.     

Sylow products and the solvable radical

We study products of Sylow subgroups of a finite group G. First we prove that G is solvable if and only if G = P₁ ... P_m for any choice of Sylow p_i-subgroups P_i , where p₁,..., p_m are all of the distinct prime divisors of |G|, and for any ordering of the p_i . Then, for a general finite group G, we show that the intersection of all Sylow products as above is a subgroup of G which is closely related to the solvable radical of G. Received: 18 November 2004     

On <Emphasis Type="Italic">c</Emphasis>-normal maximal and minimal subgroups of Sylow <Emphasis Type="Italic">p</Emphasis>-subgroups of finite groups

A subgroup H of a finite group G is called c-normal in G if there exists a normal subgroup N of G such that G = HN and $H \cap N \leq H_{G} = {\rm core}_{G}(H)$. In this paper, we investigate the class of groups of which every maximal subgroup of its Sylow p-subgroup is c-normal and the class of groups of which some minimal subgroups of its Sylow p-subgroup is c-normal for some prime number p. Some interesting results are obtained and consequently, many known results related to p-nilpotent groups and p-supersolvable groups are generalized.     

Correspondences between constituents of projective characters

Let G be a finite p-solvable group. Let P ∈ Syl _p (G) and N = N _G (P). We prove that there exists a natural bijection between the irreducible constituents of p′-degree of the principal projective character of G and those of . Received: 2 May 2007, Revised: 17 September 2007     

Finite extensions of free pro-P groups of rank at most two

For a pro-p groupG, containing a free pro-p open normal subgroup of rank at most 2, a characterization as the fundamental group of a connected graph of cyclic groups of order at mostp, and an explicit list of all such groups with trivial center are given. It is shown that any automorphism of a free pro-p group of rank 2 of coprime finite order is induced by an automorphism of the Frattini factor groupF/F ^* . Finally, a complete list of automorphisms of finite order, up to conjugacy in Aut(F), is given. Supported by an NSERC grant. Supported by the Austrian Science Foundation.     

Endomorphism algebras of transitive permutation modules for p-groups

Let G be a finite p-group with subgroup H and k a field of characteristic p. We study the endomorphism algebra E = End_kG(k_H ↑^G), showing that it is a split extension of a nilpotent ideal by the group algebra kN_G(H)/H. We identify the space of endomorphisms that factor through a projective kG-module and hence the endomorphism ring of k_H ↑^G in the stable module category, and determine the Loewy structure of E when G has nilpotency class 2 and [G, H] is cyclic. Received: 3 November 2008     

Finite groups with its power automorphism groups having small indices

In this paper, a finite group G with IAut（G） ： P（G）I ~- p or pq is determined, where P（G） is the power automorphism group of G, and p, q are distinct primes. Especially, we prove that a finite group G satisfies ｜Aut（G） ： P（G）｜= pq if and only if Aut（G）/P（G） ≌S3. Also, some other classes of finite groups are investigated and classified, which are necessary for the proof of our main results.