Semi-extraspecial $p$-groups with automorphisms of large order

Sofia Brenner; Rachel D. Camina; Mark Lewis

doi:10.5802/crmath.762

Research article - Group theory

Semi-extraspecial $p$-groups with automorphisms of large order

Sofia Brenner ¹; Rachel D. Camina ²; Mark Lewis ³

¹ Department of Mathematics, TU Darmstadt, 64289 Darmstadt, Germany
² Fitzwilliam College, Cambridge, CB3 0DG, UK
³ Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 933-940

Abstract (Original language)
Résumé

In this paper, we consider semi-extraspecial $p$-groups $G$ that have an automorphism of order $\vert G:G^{\prime } \vert - 1$. We prove that these groups are isomorphic to Sylow $p$-subgroups of $\operatorname{SU}_3 (p^{2a})$ for some integer $a$. If $p$ is odd, this is equivalent to saying that $G$ is isomorphic to a Sylow $p$-subgroup of $\operatorname{SL}_3 (p^a)$.

Dans cet article, nous considérons des $p$-groupes $G$ semi-extraspéciaux qui ont un automorphisme d’ordre $\vert G:G^{\prime } \vert - 1$. Nous prouvons que ces groupes sont isomorphes aux $p$-sous-groupes de Sylow de $\operatorname{SU}_3 (p^{2a})$ pour un certain entier $a$. Si $p$ est impair, cela revient à dire que $G$ est isomorphe à un $p$-sous-groupe de Sylow de $\operatorname{SL}_3 (p^a)$.

Received: 2025-02-04
Revised: 2025-05-31
Accepted: 2025-05-31
Published online: 2025-07-22

DOI: 10.5802/crmath.762

Classification: 20D15
Keywords: Semi-extraspecial $p$-groups, automorphisms, special unitary group
Mots-clés : Groupes $p$ semi-extraspéciaux, automorphismes, groupe unitaire spécial

Author's affiliations:

Sofia Brenner ¹; Rachel D. Camina ²; Mark Lewis ³

¹ Department of Mathematics, TU Darmstadt, 64289 Darmstadt, Germany
² Fitzwilliam College, Cambridge, CB3 0DG, UK
³ Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Sofia Brenner and Rachel D. Camina and Mark Lewis},
     title = {Semi-extraspecial $p$-groups with automorphisms of large order},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {933--940},
     year = {2025},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     doi = {10.5802/crmath.762},
     language = {en},
}

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Sofia Brenner; Rachel D. Camina; Mark Lewis. Semi-extraspecial $p$-groups with automorphisms of large order. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 933-940. doi: 10.5802/crmath.762

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