Related papers: On Group Violations of Inequalities in five Subgro…

Let $G$ be an almost simple group with socle $A_n$, the alternating group of degree $n$. We prove that there is a unit of order $pq$ in the integral group ring of $G$ if and only if there is an element of that order in $G$ provided $p$ and…

Let $p$ be any prime. We determine precisely those irreducible characters of symmetric groups which contain at most $p$ distinct linear constituents in their restriction to a Sylow $p$-subgroup, answering a question of Giannelli and…

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 finite group G is exceptional if it has a quotient Q whose minimal faithful permutation degree is greater than that of G. We say that Q is a distinguished quotient. The smallest examples of exceptional p-groups have order p^5. For an odd…

We determine the groups of minimal order in which all groups of order n can embedded for 1 < n < 16. We further determine the order of a minimal group in which all groups or order n or less can be embedded, also for 1 < n < 16.

Denote by $G$ a finite group and by $\psi(G)$ the sum of element orders in $G$. If $t$ is a positive integer, denote by $C_t$ the cyclic group of order $t$ and write $\psi(t)=\psi(C_t)$. In this paper we proved the following Theorem A: Let…

We show that if the Sylow $p$-subgroup of a finite group $G$ is of order $p$, then the normalized unit group of the integral group ring of $G$ contains a normalized unit of order $pq$ if and only if $G$ contains an element of order $pq$,…

Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\cdot...\cdot(n_kg)$ where $g\in G$ and $n_1,\cdots,n_k\in[1,{\hbox{\rm ord}}(g)]$, and the index $\ind S$ of $S$ is defined to be the minimum…

We prove several Littlewood-Offord type inequalities for arbitrary groups. In groups having elements of finite order the worst case scenario is provided by the simple random walk on a certain cyclic subgroup. The inequalities we obtain are…

A group in which every element commutes with its endomorphic images is called an $E$-group. If $p$ is a prime number, a $p$-group $G$ which is an $E$-group is called a $pE$-group. Every abelian group is obviously an $E$-group. We prove that…

Given a prime number \(p\) and a natural number \(m\) not divided by \(p\), we propose the problem of finding the smallest number \(r_{0}\) such that for \(r\geq r_{0}\), every group \(G\) of order \(p^{r}m\) has a non-trivial normal…

We study the minimal unitary representations of non-compact groups and supergroups obtained by quantization of their geometric realizations as quasi-conformal groups and supergroups. The quasi-conformal groups G leave generalized…

We show that if a finite simple group G isomorphic to PSL(n,q) or PSU(n,q), where either $n\ne 4$, or q is prime or even, acts on a vector space over a field of the defining characteristic of G, then the corresponding semidirect product…

This paper almost classifies the maximal subgroups of $E_7(q)$ for general $q$ a power of a prime $p$. Only four potential maximal subgroups are missing: $PSL_2(7)$ (unknown for $p\neq 2,3,7$), $PSL_2(8)$ ($p=2$) and $PSL_2(9)=A_6$ ($p\neq…

We give a sharp lower bound on the size of nonabelian quotients of the surface braid group $B_n(\Sigma_g)$ and classify all quotients that attain the lower bound: Depending on $n$ and $g$, a quotient of minimum order is either a symmetric…

In [3] is was shown that for any group $G$ whose rank (i.e., minimal number of generators) is at most 3, and any finite index subgroup $H\leq G$ with index $[G:H]\geq rank(G)$, one can always find a left-right transversal of $H$ which…

For a finite group $G$, denote by $\alpha(G)$ the minimum number of vertices of any graph $\Gamma$ having $\text{Aut}(\Gamma)\cong G$. In this paper, we prove that $\alpha(G)\leq |G|$, with specified exceptions. The exceptions include four…

Let $G$ be a semisimple linear algebraic group over a field $k$ and let $G^+(k)$ be the subgroup generated by the subgroups $R_u(Q)(k)$, where $Q$ ranges over all the minimal $k$-parabolic subgroups $Q$ of $G$. We prove that if $G^+(k)$ is…

We classify all of the groups with twelve or fewer subgroups. This paper is the proof of the entries in a submission to the Online Encyclopedia of Integer Sequences.

We use heuristic algorithms to find terraces for small groups. We show that Bailey's Conjecture (that all groups other than the non-cyclic elementary abelian 2-groups are terraced) holds up to order 511, except possibly at orders 256 and…