International Journal of Group Theory
Volume:14 Issue: 3, Sep 2025

Let $o(G)$ be the average order of a finite group $G$. M. Herzog, P. Longobardi and M. Maj [M. Herzog, P. Longobardi and M. Maj, Another criterion for solvability of finite groups, J. Algebra, 597 (2022) 1-23.] showed that if $G$ is non-solvable and $o(G)=o(A_5)$, then $G\cong A_5$. In this note, we prove that the equality $o(G)=o(A_5)$ does not hold for any finite solvable group $G$. Consequently, up to isomorphism,$A_5$ is determined by its average order.

In this paper, the problem existing $O$-basis for Cartesian symmetry classes is discussed. The dimensions of Cartesian symmetry classes associated with a cyclic subgroup of the symmetric group $S_m$ (generated by a product of disjoint cycles) and the product of cyclic subgroups of $S_m$ are explicitly expressed in terms of the Ramanajun sum. Additionally, a necessary and sufficient condition for the existence of an $O$-basis for Cartesian symmetry classes associated with the irreducible characters of dihedral group is given. The dimensions of these classes are also computed.

For a finite group $G$, the average order $o(G)$ is defined to be the average of all order elements in $G$, that is $o( G)=\frac{1}{|G|}\sum_{x\in G}o(x)$, where $o(x)$ is the order of element $x$ in $G$. Jaikin-Zapirain in [On the number of conjugacy classes of finite nilpotent groups, Advances in Mathematics, \textbf{227} (2011) 1129-1143] asked the following question: if $G$ is a finite ($p$-) group and $N$ is a normal (abelian) subgroup of $G$, is it true that $o(N)^{\frac{1}{2}}\leq o(G) $? We say that $G$ satisfies the average condition if $o(H)\leq o(G)$, for all subgroups $H$ of $G$. In this paer we show that every finite abelian group satisfies the average condition. This result confirms and improves the question of Jaikin-Zapirain for finite abelian groups.

According to Letourmy and Vendramin, a representation of a skew brace is a pair of representations on the same vector space, one for the additive group and the other for the multiplicative group, that satisfies a certain compatibility condition. Following their definition, we shall explain how some of the results from representation theory of groups, such as Maschke's theorem and Clifford's theorem, extend naturally to that of skew braces. We shall also give some concrete examples to illustrate that skew brace representations are more difficult to classify than group representations.

For any group $G$, define $g\sim h$ if $g,h\in G$ have the same order. The set of sizes of the equivalent classes with respect to this relation is called the same-order type of $G$. In this short note we prove that there is no finite group whose same-order type is an arithmetic progression of length $4$. This answered an open problem posed by Lazorec and  Tˇarnˇauceanu

If $G$ be a finite $p$-group and $\chi$ is a non-linear irreducible character of $G$, then $\chi(1)\leq |G/Z(G)|^{\frac{1}{2}}$. In \cite{fernandez2001groups}, Fern\'{a}ndez-Alcober and Moret\'{o} obtained the relation between the character degree set of a finite $p$-group $G$ and its normal subgroups depending on whether $|G/Z(G)|$ is a square or not. In this paper we investigate the finite $p$-group $G$ where for any normal subgroup $N$ of $G$ with $G'\not \leq N$ either $N\leq Z(G)$ or $|NZ(G)/Z(G)|\leq p$ and obtain some alternate characterizations of such groups. We find that if $G$ is a finite $p$-group with $|G/Z(G)|=p^{2n+1}$ and $G$ satisfies the condition that for any normal subgroup $N$ of $G$ either $G'\not \leq N$ or $N\leq Z(G)$, then $cd(G)=\{1, p^{n}\}$. We also find that if $G$ is a finite $p$-group with nilpotency class not equal to $3$ and $|G/Z(G)|=p^{2n}$ and $G$ satisfies the condition that for any normal subgroup $N$ of $G$ either $G'\not \leq N$ or $|NZ(G)/Z(G)|\leq p$, then $cd(G) \subseteq \{1, p^{n-1}, p^{n}\}$.