New York Journal of Mathematics
Volume 25 (2019), 949-963


Cindy (Sin Yi) Tsang

On the multiple holomorph of a finite almost simple group

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Published: September 20, 2019.
Keywords: regular subgroups, holomorph and multiple holomorph, Hopf-Galois structures, almost simple groups.
Subject: 20B35, 20D05, 08A35.

Let G be a group. Let Perm(G) denote its symmetric group and write Hol(G) for the normalizer of the subgroup of left translations in Perm(G). The multiple holomorph NHol(G) of G is defined to be the normalizer of Hol(G) in Perm(G). In this paper, we shall show that the quotient group NHol(G)/Hol(G) has order two whenever G is finite and almost simple. As an application of our techniques, we shall also develop a method to count the number of Hopf-Galois structures of isomorphic type on a finite almost simple field extension in terms of fixed point free endomorphisms.


The author would like to thank the referee for helpful comments.

Author information

Cindy (Sin Yi) Tsang:
School of Mathematics (Zhuhai)
Sun Yat-Sen University
Zhuhai, Guangdong Province, China