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.