Let N be a nilpotent group normal in a group G. Suppose that G acts transitively upon the points of a finite non-Desarguesian projective plane P. We prove that, if P has square order, then N must act semi-regularly on P. In addition we prove that if a finite non-Desarguesian projective plane P admits more than one nilpotent group which is regular on the points of P then P has non-square order a...

We prove that if p is a prime and W is the standard wreath product of two nontrivial cyclic p-groups X and Y then W is isomorphic to the full automorphism group of some group if and only if |X| = 2 and |Y | is 2 or 4.

We prove that the only primes which may divide the order of the automorphism group of a putative binary self-dual doubly-even [120, 60, 24] code are 2, 3, 5, 7, 19, 23 and 29. Furthermore we prove that automorphisms of prime order p ≥ 5 have a unique cycle structure.

We introduce conditions on a group action on a tree that are sufficient for the action to extend to the automorphism group. We apply this to two different classes of one-relator groups: certain BaumslagSolitar groups and one-relator groups with centre. In each case we derive results about the automorphism group, and deduce that there are relatively few outer automorphisms.

For each q = 2m, m ≥ 1, an infinite class of 2-(q4 + q + 1, q + 1, 1) designs admitting PSU(3, q) as an automorphism group is constructed. Such designs are shown to contain embedded Desarguesian projective planes of order q and a PSU(3, q)-invariant oval. © 2013 Elsevier B.V. All rights reserved.

We describe, up to degree n, the Lie algebra associated with the automorphism group of a free group of rank n. We compute in particular the ranks of its homogeneous components, and their structure as modules over the linear group. Along the way, we infirm (but confirm a weaker form of) a conjecture by Andreadakis, and answer a question by Bryant–Gupta–Levin– Mochizuki.

Reconstruction results give conditions under which the abstract group structure of the automorphism group Aut(M) of an ω-categorical structure M determines the topology on Aut(M), and hence determines M up to biinterpretability, by [1]; they can also give conditions under which the abstract group Aut(M) determines the permutation group 〈Aut(M),M〉, so determines M up to bi-definability. One such...

A graph X is said to be 1 2-transitive if its automorphism group Aut X acts vertex-and edge-but not arc-transitively on X. Then Aut X induces an orientation of the edges of X. If X has valency 4, then this orientation gives rise to so called alternating cycles, that is even length cycles in X whose every other vertex is the head and every other vertex is the tail of its two incident edges in th...

We show that 3 is the smallest order of a non-trivial odd order group which occurs as the full automorphism group of a finite group.