International Journal of Group Theory
Volume:7 Issue: 3, Sep 2018

ýLet G be a finite groupý. ýThe prime degree graph of Gý,denotedý ýby Δ(G) ý, ýis an undirected graph whose vertex set is ρ(G) and there is an edgeý ýbetween two distinct primes p and q if and only if pq divides some irreducibleý ýcharacter degree of G ý. ýIn generalý, ýit seems that the prime graphsý ýcontain many edges and thus they should have many trianglesý, ýso one of the cases that would be interesting is to consider those finite groups whose prime degree graphs have a small number of trianglesý. ýIn this paper we consider the case where for a nonsolvable group G ý, ýΔ(G) is a connected graph which has only one triangle and four verticesý.

In this paper, the structure of non-periodic generalized radical groups of infinite rank whose subgroups of infinite rank satisfy a suitable permutability condition is investigated.

Let G be a group and p be an endomorphism of Gý. ýA subgroup H of G is called p-inert if Hp∩H has finite index in the image Hpý. ýThe subgroups that are p-inert for all inner automorphisms of G are widely known and studied in the literatureý, ýunder the name inert subgroupsý.
ýThe related notion of inertial endomorphismý, ýnamely an endomorphism p such that all subgroups of G are p-inertý, ýwas introduced in \cite{DR1} and thoroughly studied in \cite{DR2,DR4}ý. ýThe ``dualý" ýnotion of fully inert subgroupý, ýnamely a subgroup that is p-inert for all endomorphisms of an abelian group Aý, ýwas introduced in \cite{DGSV} and further studied in \cite{Chý, ýDSZ,GSZ}ý. ýThe goal of this paper is to give an overview of up-to-date known resultsý, ýas well as some new onesý, ýand show how some applications of the concept of inert subgroup fit in the same picture even if they arise in different areas of algebraý. ýWe survey on classical and recent results on groups whose inner automorphisms are inertialý. ýMoreoverý, ýwe show howý ýinert subgroups naturally appear in the realm of locally compact topological groups or locally linearly compact topological vector spacesý, ýand can be helpful for the computation of the algebraic entropy of continuous endomorphismsý.

In this paper we consider finite groups G satisfying the followingý ýconditioný: ýG has two columns in its character table which differ by exactly oneý ýentryý. ýIt turns out that such groups exist and they are exactly the finite groupsý ýwith a non-trivial intersection of the kernels of all but one irreducibleý ýcharacters orý, ýequivalentlyý, ýfinite groups with an irreducible characterý ývanishing on all but two conjugacy classesý. ýWe investigate such groupsý ýand in particular we characterize their subclassý, ýwhich properly containsý ýall finite groups with non-linear characters of distinct degreesý, ýwhich were characterized by Berkovichý, ýChillag and Herzog in 1992ý.

Arithmetic aspects of integral representations of finite groups and their irreducibility are considered with a focus on globally irreducible representations and their generalizations to arithmetic rings. Certain problems concerning integral irreducible two-dimensional representations over number rings are discussed. Let K be a finite extension of the rational number field and OK the ring of integers of K. Let G be a finite subgroup of GL(2,K), the group of (2×2)-matrices over K. We obtain some conditions on K for G to be conjugate to a subgroup of GL(2,OK)

We address the problem of finding examples of non-bireversible transducers defining free groups, we show examples of transducers with sink accessible from every state which generate free groups, and, in general, we link this problem to the non-existence of certain words with interesting combinatorial and geometrical properties that we call fragile words. By using this notion, we exhibit a series of transducers constructed from Cayley graphs of finite groups whose defined semigroups are free, and thus having exponential growth.