Related papers: $p$-groups with $p^2$ as a codegree

Let $G$ be a finite group, $p$ a prime and $P$ a Sylow $p$-subgroup of $G$. In this note we give a cohomological criterion for the $p$-solvability of $G$ depending on the cohomology in degree $1$ with coefficients in $\mathbb F_p$ of both…

Let $G$ be a finite group and $\pi$ be a set of primes. We study finite groups with a large number of conjugacy classes of $\pi$-elements. In particular, we obtain precise lower bounds for this number in terms of the $\pi$-part of the order…

Given a group word $w$ in $k$ variables, a finite group $G$ and $g\in G$, we consider the number $N_{w,G}(g)$ of $k$-tuples $g_1,\dots ,g_k$ of elements of $G$ such that $w(g_1,\dots ,g_k)=g$. In this work we study the functions $N_{w,G}$…

Let $G$ be a group and $H \le K \le G$. We say that $H$ is $c$-embedded in $G$ with respect to $K$ if there is a subgroup $B$ of $G$ such that $G = HB$ and $H \cap B \le Z(K)$. Given a finite group $G$, a prime number $p$ and a Sylow…

Let $FG$ be the group algebra of a finite $p$-group $G$ over a finite field $F$ of positive characteristic $p$. Let $\cd$ be an involution of the algebra $FG$ which is a linear extension of an anti-automorphism of the group $G$ to $FG$. If…

Let $G$ be a finite group, and let ${\rm{cd}}(G)$ denote the set of degrees of the irreducible complex characters of $G$. Define then the character degree graph $\Delta(G)$ as the (simple undirected) graph whose vertices are the prime…

Let $G$ be a finite group and, for a prime $p$, let $S$ be a Sylow $p$-subgroup of $G$. A character $\chi$ of $G$ is called $\Syl_p$-regular if the restriction of $\chi$ to $S$ is the character of the regular representation of $S$. If, in…

An ordinary character $\chi $ of a finite group is called orthogonally stable, if all non-degenerate invariant quadratic forms on any module affording the character $\chi $ have the same discriminant. This is the orthogonal discriminant,…

Let G be a p-solvable group, P a p-subgroup and chi in Irr(G) such that chi(1)_p \ge |G:P|_p. We prove that the restriction chi_P is a sum of characters induced from subgroups Q\le P such that chi(1)_p=|G:Q|_p. This generalizes previous…

In this paper, we show that if $p$ is a prime and $G$ is a $p$-solvable group, then $| G:O_p (G) |_p \le (b(G)^p/p)^{1/(p-1)}$ where $b(G)$ is the largest character degree of $G$. If $p$ is an odd prime that is not a Mersenne prime or if…

Nonsingular derivations of modular Lie algebras which have finite multiplicative order play a role in the coclass theory for pro-$p$ groups and Lie algebras. A study of the set N_p of positive integers which occur as orders of nonsingular…

In 2000, L. H\'{e}thelyi and B. K\"{u}lshammer proved that if $p$ is a prime number dividing the order of a finite solvable group $G$, then $G$ has at least $2\sqrt{p-1}$ conjugacy classes. In this paper we show that if $p$ is large, the…

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…

Let G be a finite non-abelian simple group and let p be a prime. We classify all pairs (G,p) such that the sum of the complex irreducible character degrees of G is greater than the index of a Sylow p-subgroup of G. Our classification…

In this paper we study finite p-solvable groups having irreducible complex characters chi in Irr(G) which take roots of unity values on the p-singular elements of G.

Let $\mathcal{P}$ be a chiral polytope with type $\{k_1, k_2\}$ and $G=Aut(\mathcal{P})$. Suppose $|G|=2p^m$, where $k_1, k_2\geq 3$ and $p$ is an odd prime. Let $P$ be a Sylow $p$-subgroup of $G$. We prove that $G \cong P \rtimes…

We give a sufficient condition on a finite $p$-group $G$ of nilpotency class 2 so that $\Aut_c(G) = \Inn(G)$, where $\Aut_c(G)$ and $\Inn(G)$ denote the group of all class preserving automorphisms and inner automorphisms of $G$…

A finite group of order divisible by 3 in which centralizers of 3-elements are 3-subgroups will be called a C{\theta}{\theta}-group. The prime graph (or Gruenberg-Kegel graph) of a finite group G is denoted by {\Gamma}(G) (or GK(G)) and its…

A finite group $G$ is called {\it $p^i$-central of height $k$} if every element of order $p^i$ of $G$ is contained in the $k^{th}$-term $\zeta_k(G)$ of the ascending central series of $G$. If $p$ is odd such a group has to be $p$-nilpotent…

Let $G$ be a classical algebraic group, $X$ a maximal rank reductive subgroup and $P$ a parabolic subgroup. This paper classifies when $X\G/P$ is finite. Finiteness is proven using geometric arguments about the action of $X$ on subspaces of…