Related papers: Constructing groups of type $FP_2$ over fields but…

In this note, we establish a mass formula for self-orthogonal codes over Z_{p^2}, where p is a prime. As a consequence, an alternative proof of the known mass formulas for self-dual codes over Z_{p^2} is obtained. We also establish a mass…

We prove that for any prime number $p$, every finite non-abelian $p$-group $G$ of class 2 has a noninner automorphism of order $p$ leaving either the Frattini subgroup $\Phi(G)$ or $\Omega_1(Z(G))$ elementwise fixed.

Let $F_{q}$ be any finite field of characteristic $p>0$ having $q = p^{n}$ elements. In this paper, we have obtained the complete structure of unit groups of group algebras $F_{q}[QD_{2^k}]$, for $k = 4$ and $5$, for any prime $p>0$, where…

This paper is originally designed as a part of revision of the author's preprint math.AG/9908174 "P-adic Schwarzian triangle groups of Mumford type". Recently, Yves Andr'e pointed out a flaw in that preprint; more precisely, Proposition II…

Let $(G,Z(G))$ be a Camina pair. We prove that $G$ must be a $p$-group for some prime $p$. We also prove that $|Z(G)| < |G:Z(G)|^{3/4}$. Also, we discuss how one might build examples with $|Z(G)| > |G:Z(G)|^{1/2}$, although we are not able…

Every mathematician is familiar with the beautiful structure of finite commutative groups. What is less well known is that finite commutative semigroups also have a neat and well-described structure. We prove this in an efficient fashion.…

This note collects several results on the capability of $p$-groups of class two and prime exponent. Among the new results, we settle the 4-generator case for this class.

For $p$ a prime, $G$ a finite group and $A$ a normal subset of elements of order $p$, we prove that if $A^2 = \{ab \mid a, b \in A\}$ consists of $p$-elements then $Q = \langle A \rangle$ is soluble. Further, if $O_p(G) = 1$, we show that…

For any prime number p and any positive real number {\alpha}, we construct a finitely generated group {\Gamma} with p-gradient equal to {\alpha}. This construction is used to show that there exist uncountably many pairwise non-commensurable…

We show that all spin groups of non-definite, quinary quadratic forms over a field with characteristic 0 can be represented as 2 by 2 matrices with entries in an associated quaternion algebra. Over local and global fields, we further study…

In this paper we present new examples of simple $p$-local compact groups for all odd primes. We also develop the necessary tools to show saturation, simpleness and the non-realizability as $p$-compact groups or compact Lie groups, which can…

Let F be a non-Archimedean locally compact field of residue characteristic p, let D be a finite dimensional central division F-algebra and let l be a prime number different from p. We develop a theory of l-modular types for the group…

Given a non-isotrivial elliptic curve over $\mathbb{Q}(t)$ with large Mordell-Weil rank, we explain how one can build, for suitable small primes $p$, infinitely many fields of degree $p^2-1$ whose ideal class group has a large $p$-torsion…

We introduce a notion of highly Kummer-faithful fields and study its relationship with the notion of Kummer-faithful fields. We also give some examples of highly Kummer-faithful fields. For example, if $k$ is a number field of finite degree…

All quantum group structures on the group GL(2) are classified. It is shown that there are only two such structures, the well known quantum groups GL$_{qp}$(2) and GL$_{hh'}$(2).

In this paper we study a certain class of polycyclic groups. We outline a method for constructing a poly-$\mathbb{Z}$ group $G_n$ by describing a process for selecting maps that are used to extend $G_i$ to $G_{i+1}$ for $1 \leq i \leq n-1$…

This note discusses our formalisation in Lean of the classification of the groups of order $p q$ for (not necessarily distinct) prime numbers $p$ and $q$, together with various intermediate results such as the characterisation of internal…

We study the collection of group structures that can be realized as a group of rational points on an elliptic curve over a finite field (such groups are well known to be of rank at most two). We also study various subsets of this collection…

Given a prime $p$, we construct a permutation group containing at least $p^{p-2}$ non-conjugated regular elementary abelian subgroups of order $p^3$. This gives the first example of a permutation group with exponentially many non-conjugated…

Let $p$ be an odd prime with $p\equiv1\bmod 4$. Then for any odd power $q$ of $p$ and a positive integer $j$ we show that the groups $\text{Sp}_{p^j+1}(q),\text{PSp}_{p^j+1}(q)$, and their Sylow $p$-subgroups are non-$FSZ_{p^j}$.