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\$, egin{equation*} Hol(G) = N_{S (G)}( ho(G)) = Aut(G) ho(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 egin{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 &gt; 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 &gt; 3\$. More generally, we develop some aspects of a theory of \$T(G)\$ for \$G\$ a finite \$p\$-group of class \$2\$, for \$p &gt; 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.

Multiple holomorphs of finite p-groups of class two / Caranti, A.. - In: JOURNAL OF ALGEBRA. - ISSN 0021-8693. - STAMPA. - 516:(2018), pp. 352-372. [10.1016/j.jalgebra.2018.09.031]

### Multiple holomorphs of finite p-groups of class two

#### Abstract

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\$, egin{equation*} Hol(G) = N_{S (G)}( ho(G)) = Aut(G) ho(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 egin{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.
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2018
Caranti, A.
Multiple holomorphs of finite p-groups of class two / Caranti, A.. - In: JOURNAL OF ALGEBRA. - ISSN 0021-8693. - STAMPA. - 516:(2018), pp. 352-372. [10.1016/j.jalgebra.2018.09.031]
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11572/221847`
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