Related papers: Characters, Hall subgroups, and Normal Complements

A subgroup $H$ of a group $G$ is said to be an $IC\Phi$-subgroup of $G$ if $H \cap [H,G] \le \Phi(H)$. We analyze the structure of a finite group $G$ under the assumption that some given subgroups of $G$ are $IC\Phi$-subgroups of $G$. A new…

Following the literature, a group $G$ is called a group of central type if $G$ has an irreducible character that vanishes on $G\setminus Z(G)$. Motivated by this definition, we say that a character $\chi\in {\rm Irr}(G)$ has central type if…

Let $G\subset\GL(\BC^r)$ be a finite complex reflection group. We show that when $G$ is irreducible, apart from the exception $G=\Sgot_6$, as well as for a large class of non-irreducible groups, any automorphism of $G$ is the product of a…

Let $\chi$ be an irreducible character of a group $G.$ We denote the sum of the codegrees of the irreducible characters of $G$ by $S_c(G)=\sum_{\chi\in {\rm Irr}(G)}{\rm cod}(\chi).$ We consider the question if $S_c(G)\leq S_c(C_n)$ is true…

The codegree of an irreducible character $\chi$ of a finite group $G$ is defined as $|G:\ker\chi|/\chi(1)$. The codegree graph $\Gamma(G)$ of a finite group $G$ is the graph whose vertices are the prime divisors of $|G|$, where two distinct…

Let $G$ be a finite group. A proper subgroup $H$ of $G$ is said to be large if the order of $H$ satisfies the bound $|H|^3 \ge |G|$. In this note we determine all the large maximal subgroups of finite simple groups, and we establish an…

We classify finite-dimensional complex Hopf algebras $A$ which are pointed, that is, all of whose irreducible comodules are one-dimensional, and whose group of group-like elements $G(A)$ is abelian such that all prime divisors of the order…

We give a short proof of the fact that if all characteristic p simple modules of the finite group G have dimension less than p, then G has a normal Sylow p-subgroup.

We show that a group is a GVZ-group if and only if it is a flat group. We show that the nilpotence class of a GVZ-group is bounded by the number of distinct degrees of irreducible characters. We also show that certain CM-groups can be…

We define and study the class of inner ultrahomogeneous groups, which includes Hall's universal group and the universal locally recursively presentable group. We provide simple criteria for ample generic automorphisms, straight maximality,…

Let $H$ be an abelian subgroup of a finite group $G$ and $\pi$ the set of prime divisors of $|H|$. We prove that $|H O_{\pi}(G)/ O_{\pi}(G)|$ is bounded above by the largest character degree of $G$. A similar result is obtained when $H$ is…

We prove that if $G$ is an abelian group and $H_1x_1,\dots,H_{k}x_k$ is an irredundant (minimal) cover of $G$ with cosets, then $$|G:\bigcap_{i=1}^{k}H_{i}|=2^{O(k)}.$$ This bound is the best possible up to the constant hidden in the…

For a group $G$ and a character $\chi$ of $G$, let $c(\chi)$ denote the set of all irreducible characters of $G$, occurring in $\chi$. We prove that whenever $q\geq 8$, all non-trivial irreducible character $\chi$ of $\mathrm{PSL}_2(q)$…

It is shown that every nonsingular continuous representation of the group algebra $L^{1}(G)$ in Banach spaces is completely reducible if and only if $G$ is a compact group.

Fix a set of primes $\pi$. A finite group is said to satisfy $C_\pi$ or, in other words, to be a $C_\pi$-group, if it possesses exactly one class of conjugate $\pi$-Hall subgroups. The pronormality of $\pi$-Hall subgroups in $C_\pi$-groups…

We study the diagram alphabet of knot moves associated with the character rings of certain matrix groups. The primary object is the Hopf algebra Char-GL of characters of the finite dimensional polynomial representations of the complex group…

A closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup. In this paper we classify all irreducible subgroups of exceptional algebraic groups $G$ which are connected, closed and…

In this article we prove the following result: that for any two natural numbers k and j, and for all sufficiently large symmetric groups Sym(n), there are k disjoint sets of j irreducible characters of Sym(n), such that each set consists of…

Hecke algebras are beautiful q-extensions of Coxeter groups. In this paper, we prove several results on their characters, with an emphasis on characters induced from trivial and sign representations of parabolic subalgebras. While most of…

Let $G$ be a finite group. Let $\sigma =\{\sigma_{i} | i\in I\}$ be a partition of the set of all primes $\Bbb{P}$ and $n$ an integer. We write $\sigma (n) =\{\sigma_{i} |\sigma_{i}\cap \pi (n)\ne \emptyset \}$, $\sigma (G) =\sigma (|G|)$.…