INTERNATIONAL JOURNAL OF GROUP THEORY
Year:2013 | Volume:2 | Issue:3|Papers Count:6

Let G be a finite p-group and N be a normal subgroup of G with |N|=pn and |G/N|=pm. A result of Ellis (1998) shows that the order of the Schur multiplier of such a pair (G, N) of finite p- groups is bounded by p1/2n(2m+n-1) and hence it is equal to p1/2n(2m+n-1)-t for some non-negative integer t. Recently, the authors have characterized the structure of (G, N) when N has a complement in G and t£3. This paper is devoted to classification of pairs (G, N) when N has a normal complement in G and t=4, 5.

Let G be a group and xÎG. The cyclicizer of x is defined to be the subset Cyc(x)={yÎG| áx, yñ is cyclic}. G is said to be a tidy group if Cyc(x) is a subgroup for all xÎG. We call G to be a C-tidy group if Cyc(x) is a cyclic subgroup for all xÎG\K(G), where K (G) is the intersection of all the cyclicizers in G. In this note, we classify finite C-tidy groups with K(G)={1}.

The subgroups of symplectic groups which fix a non-zero vector of the underlying symplectic space are called affine subgroups. The split extension group A(4)@27: Sp6(2) is the affine subgroup of the symplectic group Sp8(2) of index 255. In this paper, we use the technique of the Fischer-Clifford matrices to construct the character table of the inertia group 27:O6-(2) of A(4) of index 28.

We prove that each normal automorphism of the n-periodic product of cyclic groups of odd order r³1003 is inner, whenever r divides n.

We obtain the PORC formulae for the number of non-associative algebras of dimension 2, 3 and 4 over the finite field GF(q). We also give some asymptotic bounds for the number of algebras of dimension n over GF(q).

We call H an SS-embedded subgroup of G if there exists a normal subgroup T of G such that HT is subnormal in G and HÇT£HsG, where HsG is the maximal s-permutable subgroup of G contained in H. In this paper, we investigate the influence of some SS-embedded subgroups on the structure of a finite group G. Some new results were obtained.