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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.

Huppert and Qian et al. classified finite groups for which all irreducible character degrees are consecutive. The aim of this paper is to determine the structure of finite groups whose irreducible character degrees of their proper subgroups are all consecutive.

An automorphism $\alpha$ of the group $G$ is said to be $n$-unipotent if $[g,_n\alpha]=1$ for all $g\in G$‎. ‎In this paper we obtain some results related to nilpotency of groups of $n$-unipotent automorphisms of solvable groups‎. ‎We also show that‎, ‎assuming the truth of a conjecture about the representation theory of solvable groups raised by P‎. ‎Neumann‎, ‎it is possible to produce‎, ‎for a suitable prime $p$‎, ‎an example of a f.g‎. ‎solvable group possessing a group of $p$-unipotent automorphisms which is isomorphic to an infinite Burnside group‎. ‎Conversely we show that‎, ‎if there exists a f.g‎. ‎solvable group $G$ with a non nilpotent $p$-group $H$ of $n$-automorphisms‎, ‎then there is such a counterexample where $n$ is a prime power and $H$ has finite exponent‎.

‎The Theorem 12 in [A note on‎ ‎$p$-nilpotence and solvability of finite groups‎, ‎J‎. ‎Algebra 321‎ ‎(2009) 1555--1560.] investigated the non-abelian simple groups in‎ ‎which some maximal subgroups have primes indices‎. ‎In this note we‎ ‎show that this result can be applied to prove that the finite groups‎ ‎in which every non-nilpotent maximal subgroup has prime index are‎ ‎solvable‎.

In this paper, we verify the solvability of free product of finite cyclic groups with topological methods. We use Cayley graphs and Everitt methods to construct suitable 2-complexes corresponding to the presentations of groups and their commutator subgroups. In particular, using these methods, we prove that the commutator subgroup of $mathbb{Z}_{m}*mathbb{Z}_{n}$ is free of rank $(m-1)(n-1)$ for all $m,ngeq2$

In this paper we consider finite p′-nilpotent groups which is a generalization of finite p-nilpotent groups. This generalization leads us to consider the various special subgroups such as the Frattini subgroup, Fitting subgroup, and the hypercenter in this generalized setting. The paper also considers the conditions under which product of p′-nilpotent groups will be a p′-nilpotent group.