Related papers: On finite groups whose derived subgroup has bounde…

In this paper we classify all capable finite $p$-groups with derived subgroup of order $p$ and $G/G'$ of rank $n-1$.

The Schur Theorem says that if $G$ is a group whose center $Z(G)$ has finite index $n$, then the order of the derived group $G'$ is finite and bounded by a number depending only on $n$. In the present paper we show that if $G$ is a finite…

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…

We show that an infinite residually finite boundedly generated group has an infinite chain of finite index subgroups with ranks uniformly bounded, and give (sublinear) upper bounds on the ranks of arbitrary finite index subgroups of…

For a subgroup $S$ of a group $G$, let $I_G(S)$ denote the set of commutators $[g,s]=g^{-1}g^s$, where $g\in G$ and $s\in S$, so that $[G,S]$ is the subgroup generated by $I_G(S)$. We prove that if $G$ is a $p$-soluble finite group with a…

We consider the capability of $p$-groups of class two and odd prime exponent. The question of capability is shown to be equivalent to a statement about vector spaces and linear transformations, and using the equivalence we give proofs of…

Baer characterized capable finite abelian groups (a group is capable if it is isomorphic to the quotient of some group by its center) by a condition on the size of the factors in the invariant factor decomposition (the group must be…

We show that if $G$ is any $p$-group of class at most two and exponent $p$, then there exist groups $G_1$ and $G_2$ of class two and exponent $p$ that contain $G$, neither of which can be expressed as a central product, and with $G_1$…

We consider the capability of $p$ groups of class two and odd prime exponent. We use linear algebra and counting arguments to establish a number of new results. In particular, we settle the 4-generator case, and prove a sufficient condition…

For a positive integer r we prove that if G is a profinite group in which the centralizer of every nontrivial element has rank at most r, then G is either a pro-p group or a group of finite rank. Further, if G is not virtually a pro-p…

We show that if $w$ is a multilinear commutator word and $G$ a finite group in which every metanilpotent subgroup generated by $w$-values is of rank at most $r$, then the rank of the verbal subgroup $w(G)$ is bounded in terms of $r$ and $w$…

Let $G$ be a finite group and $\alpha(G)=\frac{|C(G)|}{|G|}$\,, where $C(G)$ denotes the set of cyclic subgroups of $G$. In this short note, we prove that $\alpha(G)\leq\alpha(Z(G))$ and we describe the groups $G$ for which the equality…

Let $p>3$ be a prime. We show that if $G$ is a finite group with $p$-rank equal to 2, then $G$ involves $Qd(p)$ if and only if $G$ $p'$-involves $Qd(p)$. This allows us to use a version of Glauberman's ZJ-theorem to give a more direct…

For $G$ a finite group, let $d_2(G)$ denote the proportion of triples $(x, y, z) \in G^3$ such that $[x, y, z] = 1$. We determine the structure of finite groups $G$ such that $d_2(G)$ is bounded away from zero: if $d_2(G) \geq \epsilon >…

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.

Let $G$ be a finite group and $d$ the degree of a complex irreducible character of $G$, then write $|G|=d(d+e)$ where $e$ is a nonnegative integer. We prove that $|G|\leq e^4-e^3$ whenever $e>1$. This bound is best possible and improves on…

In 2001 Liebeck and Pyber showed that a finite simple group of Lie type is a product of $ 25 $ carefully chosen unipotent Sylow subgroups. Later, in a series of works it was shown that $ 4 $ unipotent Sylow subgroups suffice. We prove that…

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…

We show that finite quasisimple groups of Lie type in characteristic $p$ with an irreducible representation of prime degree $r$ over a finite field of characteristic $p$ have orders bounded above by a function of $r$, independent of $p$. We…

For a given m>=1, we consider the finite non-abelian groups G for which |C_G(g):<g>|<=m for every g in G\Z(G). We show that the order of G can be bounded in terms of m and the largest prime divisor of the order of G. Our approach relies on…