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Let G be a finite p-group and let F be the field of p elements. It is shown that if G is elementary abelian-by-cyclic then the isomorphism type of G is determined by FG.

On the Loewy-series of the modular group algebra of a finite p-group.

Olaf Manz, Reiner Staszewski (1986)

Journal für die reine und angewandte Mathematik

On the number of solutions of equation $x^{p^{k}} = 1$ in a finite group

Yakov Berkovich (1995)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Theorem A yields the condition under which the number of solutions of equation $x^{p^{k}} = 1$ in a finite $p$ -group is divisible by $p^{n + k}$ (here $n$ is a fixed positive integer). Theorem B which is due to Avinoam Mann generalizes the counting part of the Sylow Theorem. We show in Theorems C and D that congruences for the number of cyclic subgroups of order $p^{k}$ which are true for abelian groups hold for more general finite groups (for example for groups with abelian Sylow $p$ -subgroups).

On the $p$ -length of the product of two Shmidt groups.

Knyagina, V.N., Monakhov, V.S. (2004)

Sibirskij Matematicheskij Zhurnal

On the rarity of quasinormal subgroups

John Cossey, Stewart Stonehewer (2011)

Rendiconti del Seminario Matematico della Università di Padova

On the regular Sylow $p$ -subgroups of Chevalley groups over $Z_{p^{m}}$ .

Kolesnikov, S.G. (2006)

Sibirskij Matematicheskij Zhurnal

On the Splitting of Extensions by a Group of Prime Order.

John S. Rose (1970)

Mathematische Zeitschrift

On the structure of finite loop capable nilpotent groups

Miikka Rytty (2010)

Commentationes Mathematicae Universitatis Carolinae

In this paper we consider finite loops and discuss the problem which nilpotent groups are isomorphic to the inner mapping group of a loop. We recall some earlier results and by using connected transversals we transform the problem into a group theoretical one. We will get some new answers as we show that a nilpotent group having either $C_{p^{k}} \times C_{p^{l}}$ , $k > l \geq 0$ as the Sylow $p$ -subgroup for some odd prime $p$ or the group of quaternions as the Sylow $2$ -subgroup may not be loop capable.

On the total number of principal series of a finite abelian group.

Bentea, Lucian, Tărnăuceanu, Marius (2010)

Analele Ştiinţifice ale Universităţii “Ovidius" Constanţa. Seria: Matematică

OnCSQ-normal subgroups of finite groups

Yong Xu, Xianhua Li (2016)

Open Mathematics

We introduce a new subgroup embedding property of finite groups called CSQ-normality of subgroups. Using this subgroup property, we determine the structure of finite groups with some CSQ-normal subgroups of Sylow subgroups. As an application of our results, some recent results are generalized.

$p$ -groups, diagonalizable automorphisms and loops

Benedetto Scimemi (1971)

Rendiconti del Seminario Matematico della Università di Padova

Parallelepipeds, nilpotent groups and Gowers norms

Bernard Host, Bryna Kra (2008)

Bulletin de la Société Mathématique de France

In his proof of Szemerédi’s Theorem, Gowers introduced certain norms that are defined on a parallelepiped structure. A natural question is on which sets a parallelepiped structure (and thus a Gowers norm) can be defined. We focus on dimensions $2$ and $3$ and show when this possible, and describe a correspondence between the parallelepiped structures and nilpotent groups.

Permutable Subgroups of Infinite Groups.

Stewart E. Stonehewer (1972)

Mathematische Zeitschrift

p-Nilpotence, classifying space indecomposability, and other properties of almost all finite groups.

Hans-Werner Henn, Stewart Priddy (1994)

Commentarii mathematici Helvetici

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