Let F be a number field with ring of integers OF and let G be a finite group. We describe an approach to the study of the set of realisable classes in the locally free class group Cl(OF G) of OF G that involves applying the work of the second-named author in the context of relative algebraic K theory. For a large class of soluble groups G, including all groups of odd order, we show (subject to ...

In this paper we study the set Γn of nth-powers in certain finitely generated groups Γ. We show that, if Γ is soluble or linear, and Γn contains a finite index subgroup, then Γ is nilpotent-by-finite. We also show that, if Γ is linear and Γn has finite index (i.e. Γ may be covered by finitely many translations of Γn), then Γ is soluble-by-finite. The proof applies invariant measures on amenable...

Sufficient conditions are given for groups of finite Morley rank having non-trivial torsion-free nilpotent normal subgroups to have linear representations with small kernels. In particular, centreless connected soluble groups of finite Morley rank with torsion-free Fitting subgroups have faithful linear representations. On the way, using a notion of definable weight space, we prove that certain...

‎let $g$ be a finite group‎. ‎an element $gin g$ is called non-vanishing‎, ‎if for‎ ‎every irreducible complex character $chi$ of $g$‎, ‎$chi(g)neq 0$‎. ‎the bi-cayley graph $bcay(g,t)$ of $g$ with respect to a subset $tsubseteq g$‎, ‎is an undirected graph with‎ ‎vertex set $gtimes{1,2}$ and edge set ${{(x,1),(tx,2)}mid xin g‎, ‎ tin t}$‎. ‎let $nv(g)$ be the set‎ ‎of all non-vanishing element...

suppose $gamma$ is a graph with $v(gamma) = { 1,2, cdots, p}$and $ mathcal{f} = {gamma_1,cdots, gamma_p} $ is a family ofgraphs such that $n_j = |v(gamma_j)|$, $1 leq j leq p$. define$lambda = gamma[gamma_1,cdots, gamma_p]$ to be a graph withvertex set $ v(lambda)=bigcup_{j=1}^pv(gamma_j)$ and edge set$e(lambda)=big(bigcup_{j=1}^pe(gamma_j)big)cupbig(bigcup_{ijine(gamma)}{uv;uin v(gamma_i),vin ...

Let $G$ be a group and $H$ a subgroup of $G$‎. ‎ $H$ is said to have semi-$Pi$-property in $G$ if there is a subgroup $T$ of $G$ such that $G=HT$ and $Hcap T$ has $Pi$-property in $T$‎. ‎In this paper‎, ‎investigating on semi-$Pi$-property of subgroups‎, ‎we shall obtain some new description of finite groups‎.

the textit{commutativity degree}, $pr(g)$, of a finite group $g$ (i.e. the probability that two (randomly chosen) elements of $g$ commute with respect to its operation)) has been studied well by many authors. it is well-known that the best upper bound for $pr(g)$ is $frac{5}{8}$ for a finite non--abelian group $g$. in this paper, we will define the same concept for a finite non--abelian textit{...

‎‎in this paper we determine all finite $2$-groups of‎ ‎class $2$ in which every automorphism of order $2$ leaving the frattini subgroup elementwise fixed is inner‎.

Let n > 0 be an integer and X be a class of groups. We say that a group G satisfies the condition (X , n) whenever in every subset with n + 1 elements of G there exist distinct elements x, y such that 〈x, y〉 is in X . Let N and A be the classes of nilpotent groups and abelian groups, respectively. Here we prove that: (1) If G is a finite semi-simple group satisfying the condition (N , n), then ...