International Journal of Group Theory
Volume:2 Issue: 4, Dec 2013

Let Crn(Fp) denote the algebra of nXn circulant matrices over Fp, finite field of order p of prime characteristic p. The order of the unit groups U(Cr3(Fp)), U(Cr4(Fp)) and U(Cr5(Fp)) of algebras of circulant matrices over Fp are characterized.

Suppose that H is a subgroup of G, then H is said to be s-permutable in G, if H permutes with every Sylow subgroup of G. If HP=PH hold for every Sylow subgroup P of G with (|P|, |H|)=1), then H is called an s-semipermutable subgroup of G. In this paper, we say that H is partially S-embedded in G if G has a normal subgroup T such that HT is s-permutable in G and Hcap Tleq H_{overline{s}G}, where H_{overline{s}G} is generated by all s-semipermutable subgroups of G contained in H. We investigate the influence of some partially S-embedded minimal subgroups on the nilpotency and supersolubility of a finite group G. A series of known results in the literature are unified and generalized.

A longstanding conjecture asserts that every finite nonabelian p-group admits a noninner automorphism of order p. Let G be a finite nonabelian p-group. It is known that if G is regular or of nilpotency class 2 or the commutator subgroup of G is cyclic, or G/Z(G) is powerful, then G has a noninner automorphism of order p leaving either the center Z(G) or the Frattini subgroup Phi(G) of G elementwise fixed. In this note, we prove that the latter noninner automorphism can be chosen so that it leaves Z(G) elementwise fixed.

In this paper we have found systems of subgroups in a finite group which $Bbb P$-subnormality guarantees supersolvability of the whole group.

Let $G$ be a finite group and $d_2(G)$ denotes the probability ‎that $[x,y,y]=1$ for randomly chosen elements $x,y$ of $G$‎. ‎We ‎will obtain lower and upper bounds for $d_2(G)$ in the case where ‎the sets $E_G(x)={yin G:[y,x,x]=1}$ are subgroups of $G$ for ‎all $xin G$‎. ‎Also the given examples illustrate that all the ‎bounds are sharp.

A group G is said to be a C-tidy group if for every element x € G K(G), the set Cyc(x)={y € G | is cyclic} is a cyclic subgroup of G, where K(G) is the intersection of all the Cyc(x) in G. In this short note we determine the structure of finite C-tidy groups.

We describe under a various conditions abelian subgroups of the automorphism group $Aut(T_n)$ of the regular $n$-ary tree $T_n$, which are normalized by the $n$-ary adding machine " $tau=(e,dots, e,tau)sigma_tau$" where $sigma_tau$" is the $n$-cycle $(0, 1,dots, n-1)$. As an application, for $n=p$ a prime number, and for $n=p^2$ when $p=2$, we prove that every finitely generated soluble subgroup of $Aut(T_n)$, containing $tau$ is an extension of a torsion-free metabelian group by a finite group.