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.

The purpose of this paper is to propose an efficient method to compute the automorphism group of an arbitrary hyperelliptic function field over a given ground field of characteristic > 2 as well as over its algebraic extensions. Beside theoretical applications, knowing the automorphism group of a hyperelliptic function field also is useful in cryptography: The Jacobians of hyperelliptic curves ...

a $p$-group $g$ is $p$-central if $g^{p}le z(g)$‎, ‎and $g$ is‎ ‎$p^{2}$-abelian if $(xy)^{p^{2}}=x^{p^{2}}y^{p^{2}}$ for all $x,yin‎ ‎g$‎. ‎we prove that for $g$ a finite $p^{2}$-abelian $p$-central ‎$p$-group‎, ‎excluding certain cases‎, ‎the order of $g$ divides the ‎order of $text{aut}(g)$‎.

the coprime graph $gg$ with a finite group $g$‎ ‎as follows‎: ‎take $g$ as the vertex set of $gg$ and join two distinct‎ ‎vertices $u$ and $v$ if $(|u|,|v|)=1$‎. ‎in the paper‎, ‎we explore how the graph‎ ‎theoretical properties of $gg$ can effect on the group theoretical‎ ‎properties of $g$‎.

let $g$ be a $p$-group of order $p^n$ and $phi$=$phi(g)$ be the frattini subgroup of $g$. it is shown that the nilpotency class of $autf(g)$, the group of all automorphisms of $g$ centralizing $g/ fr(g)$, takes the maximum value $n-2$ if and only if $g$ is of maximal class. we also determine the nilpotency class of $autf(g)$ when $g$ is a finite abelian $p$-group.

We show that the automorphism group of the countable universal distributive lattice has strong uncountable cofinality, and we adapt the method to deduce the strong uncountable cofinality of the automorphism group of the countable universal generalized boolean algebra.