INTERNATIONAL JOURNAL OF GROUP THEORY
Year:2013 | Volume:2 | Issue:4|Papers Count:7

Let Crn (Fp) denote the algebra of n´n circulant matrices over Fp, the finite field of orderp a prime. The order of the unit groups U (Cr3 (Fp)), U (Cr4 (Fp)) and U (Cr5 (Fp)) of algebras of circulant matrices over Fp are computed.

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 (|Pj|, |jHj|) = 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 H  Ç T £ H s-G, where H s-G is generated by all s-semipermutable subgroups of G contained in H. We investigate the influence of some partiallyS-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 F (G) of Gelementwise xed. In this note, we prove that the latter noninner automorphism can be chosen so that it leaves Z (G) element wise fixed.

In this paper we find systems of subgroups of a finite group, which  P-subnormality guaran- tees supersolvability of the whole group.

Let G be a finite group and d2 (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 d2 (G) in the case where the sets EG (x) = {y  Î G: [y, x, x] =1} are subgroups of G for all x Î 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 Î G \ K (G), the set Cyc (x) ={y Î G | áx, y ñ is cyclic} is a cyclic subgroup of G, where K (G) = ÇxÎG Cyc (x). In this short note we determine the structure of finite C-tidy groups.

We describe under various conditions abelian subgroups of the automorphism group Aut (Tn) of the regular n-ary tree Tn, which are normalized by the n-ary adding machine t = (e, … e, t) st where st is the n-cycle (0, 1, … n - 1). As an application, for n=p a prime number, and for n=4, we prove that every soluble subgroup of Aut (Tn), containing t is an extension of a torsion-free metabelian group by a finite group.