# Darstellungstheorietage and Nikolaus Conference 2013

Speaker: David Ward (Manchester)

Title: Conjugate $$p$$-elements of Full Support that Generate the Wreath Product $$C_{p}\wr C_{p}$$

Abstract: For a symmetric group $$G:=\operatorname{Sym}(n)$$ and a conjugacy class $$X$$ of involutions in $$G$$, it is known that if the class of involutions do not have a unique fixed point, then - with a few small exceptions - given two elements $$a, x \in X$$, either $$\langle a,x \rangle$$ is isomorphic to the dihedral group $$D_8$$, or there is a further element $$y \in X$$ such that $$\langle a,y \rangle \cong \langle x,y \rangle \cong D_8$$.

The natural generalisation of this to $$p$$-elements is to consider when two conjugate $$p$$-elements generate a wreath product of two cyclic groups of order $$p$$. In this talk we give a necessary and sufficient condition for this in the case that $$n = p^2$$ and discuss the generalisation to arbitrary symmetric groups.