Related papers: On the Maximal Subgroups of $E_7(q)$ and Related A…

Let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f\geq 1$. Let $E$ be an elliptic curve with CM by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j_{K,f})$,…

For odd primes we prove some structure theorems for finite $p$-groups $G$, such that $G''\neq 1$ and $|G'/G''|=p^3$. Building on results of Blackburn and Hall, it is shown that $\lcs G3$ is a maximal subgroup of $G'$, the group $G$ has a…

The spectrum of a finite group is the set of its elements orders. Groups are said to be isospectral if their spectra coincide. For every finite simple exceptional group $L=E_7(q)$, we prove that each finite group isospectral to $L$ is…

Let $U(q)$ be a Sylow $p$-subgroup of the Chevalley groups $D_4(q)$ where $q$ is a power of a prime $p$. We describe a construction of all complex irreducible characters of $U(q)$ and obtain a classification of these irreducible characters…

We classify the nonsplit extensions of elementary abelian $p$-groups by $PSL_2(q)$, with odd $p$ dividing $q-1$, for an irreducible induced action, calculate the relevant low-dimensional cohomology groups, and describe the automorphism…

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…

Fix a finite semigroup $S$ and let $a_1,\ldots,a_k, b$ be tuples in a direct power $S^n$. The subpower membership problem (SMP) asks whether $b$ can be generated by $a_1,\ldots,a_k$. If $S$ is a finite group, then there is a folklore…

There is a conjecture that the second largest maximal cliques in Paley graphs of square order $P(q^2)$ have size $\frac{q+\epsilon}{2}$, where $q \equiv \epsilon \pmod 4$, and split into two orbits under the full group of automorphisms…

Let $G$ be a finite group and $n_p(G)$ the number of Sylow $p$-subgroups of $G$. In this paper, we prove if $n_p(G)<p^2$ then almost all numbers $n_p(G)$ are a power of a prime.

This paper gives a complete classification of the finite groups that contain a strongly closed p-subgroup for p any prime.

We study abstract finite groups with the property, called property $\hat{s}$, that all of their subrepresentations have submultiplicative spectra. Such groups are necessarily nilpotent and we focus on $p$-groups. $p$-groups with property…

In this paper we give a bound to the number of conjugacy classes of maximal subgroups of any almost simple group whose socle is a classical group of Lie type. The bound is $2n^{5.2}+n\log_2\log_2 q$, where $n$ is the dimension of the…

Given two subsets $X,Y$ of a finite group $G$, we write $\Pr(X,Y)$ for the probability that random elements $x \in X$ and $y \in Y$ commute. If $X,Y$ are subgroups, we denote by $\Pr^*(X,Y)$ the maximum real number $\epsilon$ with the…

We describe the Sylow subgroups of Gal(Q) for an odd prime p, by observing and studying their decomposition as a semidirect product of Z_p acting on F, where F is a free pro-p group, and Z_p are the p-adic integers. We determine the finite…

Up to linear transformations, we give a classification of all permutation polynomials of degree $7$ over $\mathbb{F}_{q}$ for any odd prime power $q$, with the help of the SageMath software.

The complete classification of the finite simple groups that are $(2,3)$-generated is a problem which is still open only for orthogonal groups. Here, we construct $(2, 3)$-generators for the finite odd-dimensional orthogonal groups…

We develop some new topological tools to study maximal subgroups of free idempotent generated semigroups. As an application, we show that the rank 1 component of the free idempotent generated semigroup of the biordered set of a full matrix…

We describe computer calculations that were used in 2016 to classify subgroups of the Monster isomorphic to $PSL_2(8)$, containing $7B$-elements. It turns out that there is no such $PSL_2(8)$ in the Monster. These calculations confirm…

Let $p$ be an odd prime. We construct a $p$-group $P$ of nilpotency class two, rank seven and exponent $p$, such that $\mathrm{Aut}(P)$ induces $N_{\mathrm{GL}(7,p)}(G_2(p)) = Z(\mathrm{GL}(7,p)) G_2(p)$ on the Frattini quotient…

We give here the exact maximal subgroup growth of two classes of polycyclic groups. Let $G_k = \langle x_1, x_2, ..., x_k \mid x_ix_jx_i^{-1}x_j \text{ for all } i < j \rangle$. So $G_k = \mathbb{Z} \rtimes (\mathbb{Z} \rtimes (\mathbb{Z}…