Finding involutions with small support

We show that the proportion of permutations $g$ in $S_n$ or $A_n$ such that $g$ has even order and $g^{|g|/2}$ is an involution with support of cardinality at most $\lceil n^\varepsilon \rceil$ is at least a constant multiple of $\varepsilon$. Using this result, we obtain the same conclusion for elements in a classical group of natural dimension $n$ in odd characteristic that have even order and power up to an involution with $(-1)$-eigenspace of dimension at most $\lceil n^\varepsilon \rceil$ for a linear or unitary group, or $2\lceil \lfloor n/2 \rfloor^\varepsilon \rceil$ for a symplectic or orthogonal group

Finding involutions with small support

We show that the proportion of permutations $g$ in $S_n$ or $A_n$ such that $g$ has even order and $g^{|g|/2}$ is an involution with support of cardinality at most $\lceil n^\varepsilon \rceil$ is at least a constant multiple of $\varepsilon$. Using this result, we obtain the same conclusion for elements in a classical group of natural dimension $n$ in odd characteristic that have even order and power up to an involution with $(-1)$-eigenspace of dimension at most $\lceil n^\varepsilon \rceil$ for a linear or unitary group, or $2\lceil \lfloor n/2 \rfloor^\varepsilon \rceil$ for a symplectic or orthogonal group

Elements in finite classical groups whose powers have large 1-Eigenspaces

We estimate the proportion of several classes of elements in finite classical groups which are readily recognised algorithmically, and for which some power has a large fixed point subspace and acts irreducibly on a complement of it. The estimates are used in complexity analyses of new recognition algorithms for finite classical groups in arbitrary characteristic

Identifying long cycles in finite alternating and symmetric groups acting on subsets

Let $H$ be a permutation group on a set $\Lambda$, which is permutationally isomorphic to a finite alternating or symmetric group $A_n$ or $S_n$ acting on the $k$-element subsets of points from $\{1,\ldots,n\}$, for some arbitrary but fixed $k$. Suppose moreover that no isomorphism with this action is known. We show that key elements of $H$ needed to construct such an isomorphism $\varphi$, such as those whose image under $\varphi$ is an $n$-cycle or $(n-1)$-cycle, can be recognised with high probability by the lengths of just four of their cycles in $\Lambda$.Comment: 45 page

The Divisibility Graph of finite groups of Lie Type

The Divisibility Graph of a finite group $G$ has vertex set the set of conjugacy class lengths of non-central elements in $G$ and two vertices are connected by an edge if one divides the other. We determine the connected components of the Divisibility Graph of the finite groups of Lie type in odd characteristic

On the frequency of permutations containing a long cycle

A general explicit upper bound is obtained for the proportion $P(n,m)$ of elements of order dividing $m$, where $n-1 \le m \le cn$ for some constant $c$, in the finite symmetric group $S_n$. This is used to find lower bounds for the conditional probabilities that an element of $S_n$ or $A_n$ contains an $r$-cycle, given that it satisfies an equation of the form $x^{rs}=1$ where $s\leq3$. For example, the conditional probability that an element $x$ is an $n$-cycle, given that $x^n=1$, is always greater than 2/7, and is greater than 1/2 if $n$ does not divide 24. Our results improve estimates of these conditional probabilities in earlier work of the authors with Beals, Leedham-Green and Seress, and have applications for analysing black-box recognition algorithms for the finite symmetric and alternating groups