Multiple Holomorphs of Dihedral and Quaternionic Groups

Abstract

The holomorph of a group G is Norm B G , the normalizer of the left regular representation 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 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.

Enumerating Dihedral Hopf-Galois structures acting on Dihedral extensions

“…The biggest difference is that when n is odd, any N ∈ R(D n , [D n ]) must satisfy N ≤ W (X 0 , Y 0 ) whereas if n is even, then one may have N ≤ W (X i , Y i ) for i = 0, 1, 2 potentially. As it is integral to the determination of |R(G, [G])| we examine the notion of the multiple holomorph of a group, as formulated in [7]…”

Multiple holomorphs of finite p-groups of class two

“…In all these cases, T (G) turns out to be an elementary abelian 2group. T. Kohl mentions in [Koh15] two examples where T (G) is a non-abelian 2-group; it can be verified with GAP [GAP18] that for the finite 2-groups G of order up to 8 the T (G) are elementary abelian 2groups, and that there are exactly two groups G of order 16 such that T (G) is non-abelian: for the first one T (G) is isomorphic to the dihedral group D of order 8, while for the second one T (G) is isomorphic to the direct product of D by a group of order 2. In a personal communication, Kohl has asked whether T (G) is always a 2-group when G is finite.…”

The multiple holomorph of a semidirect product of groups having coprime exponents

“…The structure of T (G) has been computed for various families of groups G; see [1][2][3][4][5][6][7][8]. In many of the known cases T (G) turns out to have order a power of 2.…”