Related papers: Some results on capable groups of prime exponent a…

We first study some families of maximal real subfields of cyclotomic fields with even class number, and then explore the implications of large plus class numbers of cyclotomic fields. We also discuss capitulation of the minus part and the…

Let G be a finitely presented group, and let p be a prime. Then G is 'large' (respectively, 'p-large') if some normal subgroup with finite index (respectively, index a power of p) admits a non-abelian free quotient. This paper provides a…

We prove a recursive formula for the exterior and symmetric powers of modules for a cyclic 2-group. This makes computation straightforward. Previously, a complete description was only known for cyclic groups of prime order.

We construct a 2-generator recursively presented group with infinite torsion length. We also explore the construction in the context of solvable and word-hyperbolic groups.

Given $g\ge 1$, the number $n(g)$ of numerical semigroups $S \subset \N$ of genus $|\N \setminus S|$ equal to $g$ is the subject of challenging conjectures of Bras-Amor\'os. In this paper, we focus on the counting function $n(g,2)$ of…

In this paper we present two new results on the number of certain conjugacy classes of a finite group. For a finite group $G$, let $n(G)$ be the maximum of $k_{p}(G)$ taken over all primes $p$ where $k_{p}(G)$ denotes the number of…

Let N and p be two prime numbers > 3 such that p divides N-1. We estimate the p-rank of the class group of Q(N^(1/p)) in terms of the discrete logarithm, with values un F_p, of certain units. Using the Gross--Koblitz formula and identities…

This article surveys modularity, level raising and level lowering questions for two-dimensional representations modulo prime powers of the absolute Galois group of the rational numbers. It contributes some new results and describes…

We classify finite $p$-groups, upto isoclinism, which have only two conjugacy class sizes $1$ and $p^3$. It turns out that the nilpotency class of such groups is $2$.

In this paper we introduce the notion of a quasi-powerful $p$-group for odd primes $p$. These are the finite $p$-groups $G$ such that $G/Z(G)$ is powerful in the sense of Lubotzky and Mann. We show that this large family of groups shares…

We sketch a simplification of proofs of old results on the arithmeticity of the group generated by opposing integral unipotent radicals in higher rank arithmetic groups

In this note, we study the finite groups whose Chermak-Delgado measure has exactly two values. They determine an interesting class of $p$-groups containing cyclic groups of prime order and extraspecial $p$-groups.

We study the finite solvable groups $G$ in which every real element has prime power order. We divide our examination into two parts: the case $\textbf{O}_2(G)>1$ and the case $\textbf{O}_2(G)=1$. Specifically we proved that if…

It is shown that the relatively free centre-by-metabelian and (class-2 nilpotent)-by-abelian pro-p groups on 2 generators are each finitely axiomatizable in the class of all profinite groups.

We present a sharp upper bound for the number of generators of a finite group in terms of the ratio between the order and the exponent.

Extending earlier work of Guralnick and of Cai and Zhang, we classify the almost simple groups which have transitive permutation representations of prime power degree $p^k$, and those which have $p$-complements (stabilisers of order coprime…

We study finite capable $p$-groups $G$ of nilpotency class 2 such that the commutator subgroup $\gamma_2(G)$ of $G$ is cyclic and the center of $G$ is contained in the Frattini subgroup of $G$.

Let $p$ be a prime divisor of the order of a finite group $G$. Then $G$ has at least $2 \sqrt{p-1}$ complex irreducible characters of degrees prime to $p$. In case $p$ is a prime with $\sqrt{p-1}$ an integer this bound is sharp for…

We investigate the structure of finite groups whose non-central real class sizes have the same $2$-part. In particular, we prove that such groups are solvable and have $2$-length one. As a consequence, we show that a finite group is…

We classify, up to conjugacy, the finite subgroups of PGL(2,K) of order prime to char(K).