International Journal of Group Theory
Volume:2 Issue: 3, Sep 2013

Let G be a finite p-group and N be a normal subgroup of G with |N|=p^n and |M|=p^m. 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 p^(1/2 n(2m+n-1)) and hence it is equal to p^(1/2 n(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 $x in G$. The cyclicizer of $x$ is defined to be the subset $Cyc(x)={y in G | is cyclic}. $G$ is said to be a tidy group if $Cyc(x)$ is a subgroup for all $x in G$. We call $G$ to be a C-tidy group if $Cyc(x)$ is a cyclic subgroup for all $x in G setminus K(G)$, where $K(G)$ is the intersection of all the cyclicizers in $G$. In this note, we classify finite C-tidy groups with $K(G)=lbrace 1 rbrace$.

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) = 2^7:SP(6,2) is the affine subgroup of the symplectic group SP(8,2) of index 255. In this paper, we use the technique of the Fischer-Clifford matrices to construct the character table of the inertia group 2^7:O-(6, 2) of A(4) of index 28.

We prove that each normal automorphism of the $n$-periodic product of cyclic groups of odd order $rge1003$ is inner، whenever $r$ divides $n$.

Graham Higman wrote two immensely important and in uential papers on enumerating p-groups in the late 1950s. The papers were entitled Enumerating p-groups I and II, and were published in the Proceedings of the London Mathematical Society in 1960. A complete description of the algebras of dimension 2 over a nite eld is given by Petersson and Scherer. They also give formulae for the number of algebras of dimension 2 over GF(q), which agree with the formulae given here. We give the formulae for the numbers of algebras of dimensions 2, 3 and 4 in Section 2. In Section 3 we give a broad outline of how the formulae can be obtained and in Section 4 we draw some quite precise conclusions about the asymptotic form of the formulae for general n.

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 Hcap Tleq H_{sG}, where H_{sG} 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. Some new results were obtained.