The Multiple Holomorphs of Finite $p$-Groups of Class Two

Abstract

$\DeclareMathOperator{\Hol}{Hol}$$\DeclareMathOperator{\Aut}{Aut}$Let $G$ be a group, and $S(G)$ be the group of permutations on the set $G$. The (abstract) holomorph of $G$ is the natural semidirect product $\Aut(G) G$. We will write $\Hol(G)$ for the normalizer of the image in $S(G)$ of the right regular representation of $G$, \begin{equation*} \Hol(G) = N_{S (G)}(ρ(G)) = \Aut(G) ρ(G) \cong \Aut(G) G, \end{equation*} and also refer to it as the holomorph of $G$. More generally, if $N$ is any regular subgroup of $S(G)$, then $N_{S(G)}(N)$ is isomorphic to the holomorph of $N$. G.A.~Miller has shown that the group \begin{equation*} T(G) = N_{S(G)}(\Hol(G))/\Hol(G) \end{equation*} acts regularly on the set of the regular subgroups $N$ of $S(G)$ which are isomorphic to $G$, and have the same holomorph as $G$, in the sense that $N_{S(G)}(N) = \Hol(G)$. If $G$ is non-abelian, inversion on $G$ yields an involution in $T(G)$. Other non-abelian regular subgroups $N$ of $S(G)$ having the same holomorph as $G$ yield (other) involutions in $T(G)$. In the cases studied in the literature, $T(G)$ turns out to be a finite $2$-group, which is often elementary abelian. In this paper we exhibit an example of a finite $p$-group $\Gp$ of class $2$, for $p>2$ a prime, which is the smallest $p$-group such that $T(\Gp)$ is non-abelian, and not a $2$-group. Moreover, $T(\Gp)$ is not generated by involutions when $p>3$. More generally, we develop some aspects of a theory of $T(G)$ for $G$ a finite $p$-group of class $2$, for $p>2$. In particular, we show that for such a group $G$ there is an element of order $p-1$ in $T(G)$, and exhibit examples where $\Size{T(G)} = p - 1$, and others where $T(G)$ contains a large elementary abelian $p$-subgroup.

19 pages This version fixes some misprints

References in corpus (3)

• Skew braces and the Yang-Baxter equation
• The multiple holomorph of a finitely generated abelian group
• Groups that have the same holomorph as a finite perfect group

Cited by in corpus (11)

• Hopf-Galois structures on extensions of degree $p^{2}q$ and skew braces of order $p^{2}q$: the cyclic Sylow $p$-subgroup case
• Bi-Skew Braces and Regular Subgroups of the Holomorph
• On bi-skew braces and brace blocks
• On the multiple holomorph of a finite almost simple group
• Brace blocks from bilinear maps and liftings of endomorphisms
• On the multiple holomorph of groups of squarefree or odd prime power order
• The multiple holomorph of a semidirect product of groups having coprime exponents
• The multiple holomorph of split metacyclic $p$-groups
• Finite $p$-groups of class two with a large multiple holomorph
• Finite $p$-groups of class two with a small multiple holomorph
• The multiple holomorph of centerless groups