‎This paper concerns aspects of various graphs whose vertex set is a group $G$‎ ‎and whose edges reflect group structure in some way (so that‎, ‎in particular‎, ‎they are invariant under the action of the automorphism group of $G$)‎. ‎The‎ ‎particular graphs I will chiefly discuss are the power graph‎, ‎enhanced power‎ ‎graph‎, ‎deep commuting graph‎, ‎commuting graph‎, ‎and non-generating graph‎. ‎My main concern is not with properties of these graphs individually‎, ‎but ‎‎‎‎rather with comparisons between them‎. ‎The graphs mentioned‎, ‎together‎ ‎with the null and complete graphs‎, ‎form a hierarchy (as long as $G$ is‎ ‎non-abelian)‎, ‎in the sense that the edge set of any one is contained in that‎ ‎of the next; interesting questions involve when two graphs in the hierarchy‎ ‎are equal‎, ‎or what properties the difference between them has‎. ‎I also ‎‎‎consider various properties such as universality and forbidden subgraphs‎, ‎comparing how these properties play out in the different graphs‎. ‎I have also included some results on intersection graphs of subgroups of‎ ‎various types‎, ‎which are often in a ``dual'' relation to one of the other‎ ‎graphs considered‎. ‎Another actor is the Gruenberg--Kegel graph‎, ‎or prime graph‎, ‎of a group‎: ‎this very small graph has a surprising influence over various‎ ‎graphs defined on the group‎. ‎Other graphs which have been proposed‎, ‎such as the nilpotence‎, ‎solvability‎, ‎and Engel graphs‎, ‎will be touched on rather more briefly‎. ‎My emphasis is on‎ ‎finite groups but there is a short section on results for infinite groups‎. ‎There are briefer discussions of general $Aut(G)$-invariant graphs‎, ‎and structures other than groups (such as semigroups and rings)‎. ‎Proofs‎, ‎or proof sketches‎, ‎of known results have been included where possible‎. ‎Also‎, ‎many open questions are stated‎, ‎in the hope of stimulating further‎ ‎investigation‎.

A subgroup $A$ of a group $G$ is called {it seminormal} in $G$‎, ‎if there exists a subgroup $B$ such that $G=AB$ and $AX$~is a subgroup of $G$ for every‎ ‎subgroup $X$ of $B$‎. ‎The group $G = G_1 G_2 cdots G_n$ with pairwise permutable subgroups $G_1‎,‎ldots‎,‎G_n$ such that $G_i$ and $G_j$ are seminormal in~$G_iG_j$ for any $i‎, ‎jin {1,ldots‎,‎n}$‎, ‎$ineq j$‎, ‎is studied‎. ‎In particular‎, ‎we prove that if $G_iin frak F$ for all $i$‎, ‎then $G^frak Fleq (G^prime)^frak N$‎, ‎where $frak F$ is a saturated formation and $frak U subseteq frak F$‎. ‎Here $frak N$ and $frak U$‎~ ‎are the formations of all nilpotent and supersoluble groups respectively‎, ‎the $mathfrak F$-residual $G^frak F$ of $G$ is the intersection of all those normal‎ ‎subgroups $N$ of $G$ for which $G/N in mathfrak F$‎.

There is a long-standing conjecture attributed to I. Schur that if $G$ is a finite group with Schur multiplier $M(G)$ then the exponent of $M(G)$ divides the exponent of $G$. In this note I give an example of a four generator group $G$ of order $5^{4122}$ with exponent $5$, where the Schur multiplier $M(G)$ has exponent $25$.

‎We show that for each positive integer $n$‎, ‎there exist a group $G$ and a subgroup $H$ such that the ordinary depth $d(H‎, ‎G)$ is $2n$‎. ‎This solves the open problem posed by Lars Kadison whether even ordinary depth larger than $6$ can occur‎.

‎Let $V$ be a unitary space‎. ‎Suppose $G$ is a subgroup of the symmetric group of degree $m$ and $\Lambda$ is an irreducible unitary representation of $G$ over a vector space $U$‎. ‎Consider the generalized symmetrizer on the tensor space $U\otimes V^{\otimes m}$‎, ‎$$ S_{\Lambda}(u\otimes v^{\otimes})=\dfrac{1}{|G|}\sum_{\sigma\in G}\Lambda(\sigma)u\otimes v_{\sigma^{-1}(1)}\otimes\cdots\otimes v_{\sigma^{-1}(m)} $$ defined by $G$ and $\Lambda$‎. ‎The image of $U\otimes V^{\otimes m}$ under the map $S_\Lambda$ is called the generalized symmetry class of tensors associated with $G$ and $\Lambda$ and is denoted by $V_\Lambda(G)$‎. ‎The elements in $V_\Lambda(G)$ of the form $S_{\Lambda}(u\otimes v^{\otimes})$ are called generalized decomposable tensors and are denoted by $u\circledast v^{\circledast}$‎. ‎For any linear operator $T$ acting on $V$‎, ‎there is a unique induced operator $K_{\Lambda}(T)$ acting on $V_{\Lambda}(G)$ satisfying $$ K_{\Lambda}(T)(u\otimes v^{\otimes})=u\circledast Tv_{1}\circledast \cdots \circledast Tv_{m}‎. ‎$$ If $\dim U=1$‎, ‎then $K_{\Lambda}(T)$ reduces to $K_{\lambda}(T)$‎, ‎induced operator on symmetry class of tensors $V_{\lambda}(G)$‎. ‎In this paper‎, ‎the basic properties of the induced operator $K_{\Lambda}(T)$ are studied‎. ‎Also some well-known results on the classical Schur functions will be extended to the case of generalized Schur functions‎.

‎Let $G$ be a finite group which is not cyclic of prime power order‎. ‎The join graph $Delta(G)$ of $G$ is a graph whose vertex set is the set of all proper subgroups of $G$‎, ‎which are not contained in the Frattini subgroup $G$ and two distinct vertices $H$ and $K$ are adjacent if and only if $G=langle H‎, ‎Krangle$‎. ‎Among other results‎, ‎we show that if $G$ is a finite cyclic group and $H$ is a finite group such that $Delta(G)congDelta(H)$‎, ‎then $H$ is cyclic‎. ‎Also we prove that $Delta(G)congDelta(A_5)$ if and only if $Gcong A_5$‎.

Let R be a non trivial finite commutative ring with identity and G be a non trivial group. We denote by P(RG) the probability that the product of two randomly chosen elements of a finite group ring RG is zero. We show that P(RG) <0.25 if and only if RG is not isomorphic to Z2C2, Z3C2, Z2C3. Furthermore, we give the upper bound and lower bound for P(RG). In particular, we present the general formula for P(RG), where R is a finite field of characteristic p and |G| ≤ 4.

We study the automorphism groups of finite-dimensional cyclic Leibniz algebras. In this connection, we consider the relationships between groups, modules over associative rings and Leibniz algebras.

The co-maximal subgroup graph $\Gamma(G)$ of a group $G$ is a graph whose vertices are non-trivial proper subgroups of $G$ and two vertices $H$ and $K$ are adjacent if $HK = G$‎. ‎In this paper‎, ‎we study and characterize various properties like diameter‎, ‎domination number‎, ‎perfectness‎, ‎hamiltonicity‎, ‎etc‎. ‎of $\Gamma(\mathbb{Z}_n)$