Enumeration of skew morphisms of cyclic $2$-groups

Abstract

A skew morphism of a finite group $B$ is a permutation of $B$ fixing the identity and satisfying $\phi (xy)=\phi (x){\phi }^{i_x}(y)$ for some integers $i_x$ indexed by $x\in B$. The enumeration of skew morphisms of finite cyclic groups remains an open problem. The most substantial progress to date concerns cyclic $p$-groups with $p$ odd, for which a full classification and enumeration was obtained by Kovács and Nedela. In this paper we treat the remaining case $p=2$, giving a complete classification and enumeration of skew morphisms of finite cyclic $2$-groups. Writing $\mathrm{Skew}(n)$ for the number of skew morphisms of ${\mathbb{Z}}_n$, we prove that $\mathrm{Skew}(2^e)=4\mathrm{Skew}(2^{e-1})-4$ for each $e\ge 4$, and that $\mathrm{Skew}(2^e)=(7\cdot 4^{e-2}+8)/6$ for each $e\ge 3$. This completes the enumeration of skew morphisms for all cyclic $p$-groups.

15 pages, 1 table