**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.