Related papers: Second maximal subgroups of the finite alternating…

M.Newman has asked if it is the case that whenever H and K are isomorphic subgroups of a finite solvable group G with H maximal, then K is also maximal. This question was considered in a paper of I.M. Isaacs and the second author, where…

Two extremal classes of acyclic groups are discussed. For an arbitrary group G, there is always a homomorphism from an acyclic group of cohomological dimension 2 onto the maximum perfect subgroup of G, and there is always an embedding of G…

We determine the number of elements of order two in the group of normalized units V(F_2G) of the group algebra F_2G of a 2-group of maximal class over the field F_2 of two elements. As a consequence for the 2-groups G and H of maximal class…

For each finite classical group $G$, we classify the subgroups of $G$ which act transitively on a $G$-invariant set of subspaces of the natural module, where the subspaces are either totally isotropic or nondegenerate. Our proof uses the…

Given a subset $W$ of an abelian group $G$, a subset $C$ is called an additive complement for $W$ if $W+C=G$; if, moreover, no proper subset of $C$ has this property, then we say that $C$ is a minimal complement for $W$. It is natural to…

We describe simply connected compact exceptional simple Lie groups in very elementary way. We first construct all simply connected compact exceptional Lie groups G concretely. Next, we find all involutive automorphisms of G, and determine…

We revisit the results on admissible transformations between normal linear systems of second-order ordinary differential equations with an arbitrary number of dependent variables under several appropriate gauges of the arbitrary elements…

Let $G$ be finite group. A subgroup $H$ of $G$ is said to be an $SS$-quasinormal subgroup of $G$, if there exists a subgroup $B$ of $G$ such that $G = HB$ and $H$ permutes with every Sylow subgroup of $B$. Let $\Omega:…

Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $\sigma $-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall…

We determine all maximal subgroups of the almost simple groups with socle $T=\PSL(2,q)$, that is, of all groups $G$ such that $\PSL(2,q)\leqslant G\leqslant\PGammaL(2,q)$, with $q\geq 4$.

We determine all finite subgroups of simple algebraic groups that have irreducible centralizers - that is, centralizers whose connected component does not lie in a parabolic subgroup.

A subgroup $H$ of a group $G$ is said to be pronormal in $G$ if $H$ and $H^g$ are conjugate in $\langle H, H^g \rangle$ for every $g \in G$. In this paper we classify finite simple groups $E_6(q)$ and ${}^2E_6(q)$ in which all the subgroups…

A primary covering of a finite group $G$ is a family of proper subgroups of $G$ whose union contains the set of elements of $G$ having order a prime power. We denote with $\sigma_0(G)$ the smallest size of a primary covering of $G$, and…

The normal covering number $\gamma(G)$ of a finite, non-cyclic group $G$ is the minimum number of proper subgroups such that each element of $G$ lies in some conjugate of one of these subgroups. We find lower bounds linear in $n$ for…

Answering a question of A. V. Vasil'ev, we show that each finite symmetric (or alternating) group $H$ is a retract of any group containing $H$ as a verbally closed subgroup.

We classify the locally compact second-countable (l.c.s.c.) groups $A$ that are abelian and topologically characteristically simple. All such groups $A$ occur as the monolith of some soluble l.c.s.c. group $G$ of derived length at most $3$;…

We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense,…

Given an arbitrary group $G$ we construct a semigroup of idempotents (band) $B_G$ with the property that the free idempotent generated semigroup over $B_G$ has a maximal subgroup isomorphic to $G$. If $G$ is finitely presented then $B_G$ is…

A group $G$ is said to have dense solitary subgroups if each non-empty open interval in its subgroup lattice $L(G)$ contains a solitary subgroup. In this short note, we find all finite groups satisfying this property.

Recent results of Qu and Tuarnauceanu describe explicitly the finite p-groups which are not elementary abelian and have the property that the number of their subgroups is maximal among p-groups of a given order. We complement these results…