Related papers: Finite solvable tidy Groups whose orders are divis…

In this article we prove results about finite soluble groups that act with fixity 2 or 3.

We construct two finite groups of size $2^{365}\cdot 3^{105}\cdot 7^{104}$: a solvable group $G$ and a non-solvable group $H$, such that for every integer $n$ the groups have the same number of elements of order $n$. This answers a question…

In this note we study the finite groups whose subgroup lattices are dismantlable.

In this paper we prove that free solvable groups have finite Krull dimension. In fact, this is true for much wider class of solvable groups, termed rigid groups. Along the way we study the algebraic structure of the limit solvable groups…

Given a prime power $p^d$ with $p$ a prime and $d$ a positive integer, we classify the finite groups $G$ with $p^{2d}$ dividing $|G|$ in which all subgroups of order $p^d$ are complemented and the finite groups $G$ having a normal…

Let $\lambda(G)$ be the maximum number of subgroups in an irredundant covering of a finite group $G$. We prove that the finite groups with $\lambda(G)=|G|-t$, where $t\leq 5$, are solvable, and classify such groups.

For a finite group $G$, the prime graph $\Gamma(G)$ (also known as Gruenberg-Kegel graph) is defined to be the graph where the vertices are the primes that divide $|G|$ such that two vertices $p$ and $q$ share an edge if and only if there…

A group is said to be cube-free if its order is not divisible by the cube of any prime. Let $f_{cf,sol}(n)$ denote the isomorphism classes of solvable cube-free groups of order $n$. We find asymptotic bounds for $f_{cf,sol}(n)$ in this…

In this paper we are concerned with finite soluble groups $G$ admitting a factorisation $G=AB$, with $A$ and $B$ proper subgroups having coprime order. We are interested in bounding the Fitting height of $G$ in terms of some…

Two finite groups are said to have the same order type if for each positive integer $n$ both groups have the same number of elements of order $n$. In 1987 John G. Thompson asked if in this case the solvability of one group implies the…

Let $\pi$ be a set of primes containing $2$ and an odd prime $p$. It is proved that if a finite group $G$ has a Hall $\pi$-subgroup $H$, then the non-$p$-soluble length of $G$ is bounded above by the generalized Fitting height of $H$. The…

In this paper, we show that each finite group $G$ containing at most $p^2$ Sylow $p$-subgroups for each odd prime number $p$, is a solvable group. In fact, we give a positive answer to the conjecture in \cite{Rob}.

We provide polynomial lower bounds for residual finiteness of residually finite, finitely generated solvable groups that admit infinite order elements in the Fitting subgroup of strict distortion at least exponential. For this class of…

Let $p$ be a prime number and suppose that every maximal subgroup of a finite group is either $p$-nilpotent or has prime index. Such group need not be $p$-solvable, and we study its structure by proving that only one nonabelian simple group…

The `upper rank' of a group is the supremum of the (Pr\"{u}fer) ranks of its finite quotients, and for a prime $p$, the `upper $p$-rank' is the supremum of the sectional $p$-ranks of those quotients. The former is finite if and only if the…

Two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. Consider the map which sends (the isomorphism class of) an S-arithmetic group to (the isomorphism class of) its…

In this paper, we study the structures of finite groups using some arithmetic conditions on the sizes of real conjugacy classes. We prove that a finite group is solvable if the prime graph on the real class sizes of the group is…

Let $G$ be a finite group and $p$ a fixed prime divisor of $|G|$. Combining the nilpotence, the normality and the order of groups together, we prove that if every maximal subgroup of $G$ is nilpotent or normal or has $p'$-order, then (1)…

A group is said to be capable if it is the central factor of some group. In this paper, among other results we have characterized capable groups of order $p^2q$, for any distinct primes $p, q$, which extends Theorem 1.2 of S. Rashid, N. H.…

We prove that certain Fuchsian triangle groups are profinitely rigid in the absolute sense, i.e. each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. We also develop a method…