Related papers: ${\rm B}_\pi$-characters and quotients

Let $U_n$ denote the group of $n\times n$ unipotent upper-triangular matrices over a fixed finite field $\FF_q$, and let $U_\cP$ denote the pattern subgroup of $U_n$ corresponding to the poset $\cP$. This work examines the superclasses and…

Let G be an isotropic reductive algebraic group over a commutative ring R. Assume that the elementary subgroup E(R) of group of points G(R) is correctly defined. Then E(R) is perfect, except for the well-known cases of a split reductive…

For G a group definable in an NIP theory we prove that there is a smallest type-definable subgroup H of G such that the quotient G/H is stable. This generalizes the existence of G^00, the smallest type-definable subgroup of G of bounded…

Let $K/k$ be a finite Galois extension and $\pi = \fn{Gal}(K/k)$. An algebraic torus $T$ defined over $k$ is called a $\pi$-torus if $T\times_{\fn{Spec}(k)} \fn{Spec}(K)\simeq \bm{G}_{m,K}^n$ for some integer $n$. The set of all algebraic…

A complex irreducible character of a finite group G is said to be p-constant, for some prime p dividing the order of G, if it takes constant value at the set of p-singular elements of G. In this paper we classify irreducible p-constant…

Let $p$ be a prime and $G$ a subgroup of $GL_d(p)$. We define $G$ to be $p$-exceptional if it has order divisible by $p$, but all its orbits on vectors have size coprime to $p$. We obtain a classification of $p$-exceptional linear groups.…

We calculate the fundamental groups $\pi=\pi_1(P^2\setminus B)$ for all irreducible plane sextics $B\subset\P^2$ with simple singularities for which $\pi$ is known to admit a dihedral quotient $D_{10}$. All groups found are shown to be…

In this paper we describe some properties of groups $G$ that contain a solvable subgroup of finite prime-power index (Theorem 1 and Corollaries 2--3). We prove that if $G$ is a non-solvable group that contains a solvable subgroup of index…

We first give a short group theoretic proof of the following result of Lackenby. If $G$ is a large group, $H$ is a finite index subgroup of $G$ admitting an epimorphism onto a non--cyclic free group, and $g$ is an element of $H$, then the…

For a finite group $G$, let $d(G)$ denote the probability that a randomly chosen pair of elements of $G$ commute. We prove that if $d(G)>1/s$ for some integer $s>1$ and $G$ splits over an abelian normal nontrivial subgroup $N$, then $G$ has…

The proof of the inductive McKay condition has been shown to imply that the character theory above the characters of degree not divisible by $p$ of a normal subgroup is locally determined. In this note, we establish a similar result for the…

In this note we give a particularly short and simple proof of the following theorem of Karrass and Solitar. Let $H$ be a finitely generated subgroup of a free group $F$ with infinite index $[F:H]$. Then there is a nontrivial normal subgroup…

Given a map with underlying graph $\mathcal{G}$, if the set of prime divisors of $|V(\mathcal{G}|$ is denoted by $\pi$, then we call the map a {\it $\pi$-map}. An orientably-regular (resp. A regular ) $\pi$-map is called {\it solvable} if…

Let $G$ be a finite solvable group, and let $p$ be a prime. In this note, we prove that $p$ does not divide $\varphi(1)$ for every irreducible monomial $p$-Brauer character $\varphi$ of $G$ if and only if $G$ has a normal Sylow…

Let the group $G = AB$ be the product of the subgroups $A$ and $B$. We determine some structural properties of $G$ when the $p$-elements in $A\cup B$ have prime power indices in $G$, for some prime $p$. More generally, we also consider the…

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…

We prove that if $b$ is a block of a finite group with normal abelian defect group and inertial quotient a direct product of elementary abelian groups, then $\operatorname{Picent}(b)$ is trivial. We also provide examples of blocks $b$ of…

We prove that if $\pi$ is a recursive set of primes, then pointlike sets are decidable for the pseudovariety of semigroups whose subgroups are $\pi$-groups. In particular, when $\pi$ is the empty set, we obtain Henckell's decidability of…

We introduce a class of finite semigroups obtained by considering Rees quotients of numerical semigroups. Several natural questions concerning this class, as well as particular subclasses obtained by considering some special ideals, are…

Let $G$ be a group, $R$ be a $G$-graded commutative ring with nonzero unity and $GI(R)$ be the set of all graded ideals of $R$. Suppose that $\phi:GI(R)\rightarrow GI(R)\cup\{\emptyset\}$ is a function. In this article, we introduce and…