Multiple Holomorphs of Dihedral and Quaternionic Groups

Abstract

The holomorph of a group G is Norm _B (λ(G)), the normalizer of the left regular representation λ(G) in its group of permutations B = Perm(G). The multiple holomorph of G is the normalizer of the holomorph in B. The multiple holomorph and its quotient by the holomorph encodes a great deal of information about the holomorph itself and about the group λ(G) and its conjugates within the holomorph. We explore the multiple holomorphs of the dihedral groups D _n and quaternionic (dicyclic) groups Q _n for n ≥ 3.

Key Words:

• Holomorph
• Regular subgroup

2010 Mathematics Subject Classification:

• 20B35
• 20F28

Notes

Communicated by D. Macpherson.