Narrow Results

This paper continues the investigation of the behavior of the commutator subgroup of an uncountable group whose large subgroups have finite (bounded) index in their normal closures. Moreover, uncountable groups in which all large subgroups have small index in their normal closures are considered.

We investigate when discrete, amenable groups have $C^{\ast }$-algebras of real rank zero. While it is known that this happens when the group is locally finite, the converse is an open problem. We show that if $C^{\ast }(G)$ has real rank zero, then all normal subgroups of $G$ that are elementary amenable and have finite Hirsch length must be locally finite.

Birget, Margolis, Meakin and Weil proved that a finitely generated subgroup K of a free group is pure if and only if the transition monoid M (K) of its Stallings automaton is aperiodic. In this paper, we establish further connections between algebraic properties of K and algebraic properties of M (K). We mainly focus on the cases where M (K) belongs to the pseudovariety of finite monoids all of whose subgroups lie in a given pseudovariety of finite groups. We also discuss normal, malnormal and cyclonormal subgroups of $F_A$ using the transition monoid of the corresponding Stallings automaton.

Let G be a finite group and p a prime. We denote by ${\mathrm{\text{Irr}}}_p(G)$ the set of irreducible complex characters of G whose degrees are linear or divisible by p, and we write ${\mathrm{\text{R}}}_p(G)$ to denote the ratio of the sum of squares of irreducible character degrees in ${\mathrm{\text{Irr}}}_p(G)$ to the sum of irreducible character degrees in ${\mathrm{\text{Irr}}}_p(G)$. The Itô–Michler Theorem on character degrees states that ${\mathrm{\text{R}}}_p(G)=1$ if and only if G has a normal abelian Sylow p-subgroup. We generalize this theorem as follows: if ${\mathrm{\text{R}}}_p(G)<\frac{p+1}{2}$, then G has a normal Sylow p-subgroup.

In this paper, we prove that a finite group G is isomorphic to the alternating group A_n with n≥5 if and only if they have the same set of the orders of solvable subgroups.

It is known that there exist soluble groups of infinite rank which satisfy the minimal condition on normal subgroups. We prove here that if G is any soluble group satisfying the minimal condition on normal subgroups of infinite rank, then either G has finite rank or it satisfies the minimal condition on normal subgroups.

In this paper, we prove that a group is nilpotent of class at most two if and only if each of its self-centralizing subgroups is normal. We also show that if all self-centralizing subgroups of a certain group are subnormal then all subgroups are subnormal.

Continuous-time quantum walks (CTQW) over finite group schemes is investigated, where it is shown that some properties of a CTQW over a group scheme defined on a finite group G induces a CTQW over group scheme defined on G/H, where H is a normal subgroup of G with prime index. This reduction can be helpful in analyzing CTQW on underlying graphs of group schemes. Even though this claim is proved for normal subgroups with prime index (using the Clifford's theorem from representation theory), it is checked in some examples that for other normal subgroups or even non-normal subgroups, the result is also true! It means that CTQW over the graph on G, starting from any arbitrary vertex, is isomorphic to the CTQW over the quotient graph on G/H if we take the sum of the amplitudes corresponding to the vertices belonging to the same cosets.

Let G be a finite group. We say that a subgroup H of G is -normal in G if G has a subnormal subgroup T such that TH = G and (H ∩ T)H_G/H_G is contained in the -hypercenter of G/H_G, where is the class of the finite supersoluble groups. We study the structure of G under the assumption that some subgroups of G are -normal in G.

Let $G$ be a finite group and $N$ a normal subgroup of $G$. Denote by ${\Gamma }_G(N)$ the graph whose vertices are all distinct $G$-conjugacy class sizes of non-central elements in $N$, and two vertices of ${\Gamma }_G(N)$ are adjacent if and only if they are not coprime numbers. We prove that if the center $Z(N)=Z(G)\cap N$ and ${\Gamma }_G(N)$is $k$-regular for $k\ge 1$, then either a section of $N$is a quasi-Frobenius group or ${\Gamma }_G(N)$ is a complete graph with $k+1$ vertices.

A group is SimpHAtic if it acts geometrically on a simply connected simplicially hereditarily aspherical (SimpHAtic) complex. We show that finitely presented normal subgroups of the SimpHAtic groups are either:

• finite, or
• of finite index, or
• virtually free.

This result applies, in particular, to normal subgroups of systolic groups. We prove similar strong restrictions on group extensions for other classes of asymptotically aspherical groups.

The proof relies on studying homotopy types at infinity of groups in question.

We also show that non-uniform lattices in SimpHAtic complexes (and in more general complexes) are not finitely presentable and that finitely presented groups acting properly on such complexes act geometrically on SimpHAtic complexes. In Appendix we present the topological two-dimensional quasi-Helly property of systolic complexes.