Let $G$ be a group and $S$ be the collection of all non-trivial proper subgroups of $G$. The co-maximal subgroup graph $\Gamma(G)$ of a group $G$ is defined to be a graph with $S$ as the set of vertices and two distinct vertices $H$ and $K$ are adjacent if and only if $HK=G$. In this paper, we study the comaximal subgroup graph on finite dihedral groups. In particular, we study order, maximum and minimum degree, diameter, girth, domination number, chromatic number and perfectness of comaximal subgroup graph of dihedral groups. Moreover, we prove some isomorphism results on comaximal subgroup graph of dihedral groups.

This paper is dealing with a split extension group of the form 26: (3,A5),which is the largest maximal subgroup of the Symplectic group Sp(4,4): We refer to this extension by G: We , rstly determine the conjugacy classes of G using the coset analysis technique. The structures of inertia factor groups were determined. We then compute the Fischer matrices of G and apply the Cli, ord-Fischer theory to calculate the ordinary character table of this group. The Fischer matrices of G are all integer valued, with sizes ranging from 1 to 4. The full character table of G is 26, 26 complex valued matrix and is given at the end of this paper.

The Mathieu group M24 has a maximal subgroup of the form G ̅=N:G, where N=26 and G=3. S6 ≅ 3. PGL2 (9). Using Atlas, we can see that M24 has only one maximal subgroup of type 26:(3. S6). The group is a split extension of an elementary abelian group, N=26 by a non-split extensionmgroup, G=3. S6. The Fischer matrices for each class representative of G are computed which together with character tables of inertia factor groups of G lead to the full character table of G ̅. The complete fusion of G ̅ into the parent group M24 has been determined using the technique of set intersections of characters.

The non-split extension group `G=53. L (3, 5) is a subgroup of order 46500000 and of index 1113229656 in Ly. The group `G in turn has L (3, 5) and 52: 2: A5 as inertia factors. The group 52: 2: A5 is of order 3000 and is of index 124 in L (3, 5). The aim of this paper is to compute the Fischer-Cliord matrices of `G, which together with associated partial character tables of the inertia factor groups, are used to compute a full character table of `G. A partial projective character table corresponding to 52: 2A5 is required, hence we have to compute the Schur multiplier and projective character table of 52: 2A5.

Let F be a field and M be a maximal subgroup of the multiplicative group F*= F \ {0} of index p. It is proved that if M is divisible, then Br(F)p ¹0 if and only if p = 2 and F is Euclidean. Furthermore, it is shown that in this case F* contains a divisible maximal subgroup if and only if F* is isomorphic to the multiplicative group of a real closed field.

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.

In 1970, Menegazzo [Gruppi nei quali ogni sottogruppo e intersezione di sottogruppi massimali, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 48 (1970), 559 {562.] gave a complete description of the structure of soluble IM -groups, i.e., groups in which every subgroup can be obtained as intersection of maximal subgroups. A group G is said to have the FM -property if every subgroup of G has finite index in the intersection xÙ of all maximal subgroups of G containing X. The behaviour of (generalized) soluble FM -groups is studied in this paper. Among other results, it is proved that ifG is a (generalized) soluble group for which there exists a positive integerk such that |xÙ:X|£k for each subgroup X, then G is finite-by- IM -by-finite, i.e., G contains a finite normal subgroup N such that G/N is a finite extension of an IM -group.