International Journal of Group Theory
Volume:1 Issue: 4, Dec 2012

A famous theorem of Schur states that for a group G finiteness of G/Z(G) implies the finiteness of G′. The converse of Schur’s theorem is an interesting problem which has been considered by some authors. Recently, Podoski and Szegedy proved the truth of the converse of Schur’s theorem for capable groups. They also established an explicit bound for the index of the center of such groups. This paper is devoted to determine some families of groups among non-capable groups which satisfy the converse of Schur’s theorem and at the same time admit the Podoski and Szegedy’s bound as the upper bound for the index of their centers.

In their paper «Finite groups whose noncentral commuting elements have centralizers of equal size», S. Dolfi, M. Herzog and E. Jabara classify the groups in question- which they call CH-groups- up to finite p-groups. Our goal is to investigate the finite p-groups in the class. The chief result is that a finite p-group that is a CH-group either has an abelian maximal subgroup or is of class at most p+1. Detailed descriptions, in some cases characterisations up to isoclinism, are given.

The aim of this paper is to classify the finite simple groups with the number of zeros at most seven greater than the number of nonlinear irreducible characters in the character tables. We find that they are exactly A$_{5}$, L$_{2}(7)$ and A$_{6}$.

In this paper we consider the group algebra R(C_2 ×D_infinity). It is shown that R(C_2 ×D_infinity) can be represented by a 4 × 4 block circulant matrix. It is also shown that U(Z_2(C_2 × D_infinity)) is nfinitely generated.

In cite{Demp2} Dempwolff proved the existence of a group of the form $2^{5}{^{cdot}}GL(5,2)$ (a non split extension of the elementary abelian group $2^{5}$ by the general linear group $GL(5,2)$). This group is the second largest maximal subgroup of the sporadic Thompson simple group $mathrm{Th}.$ In this paper we calculate the Fischer matrices of Dempwolff group $overline{G} =2^{5}{^{cdot}}GL(5,2).$ The theory of projective characters is involved and we have computed the Schur multiplier together with a projective character table of an inertia factor group. The full character table of $overline{G}$ is then can be calculated easily.

We investigate Graham Higman''s paper Enumerating p-groups, II, in whichhe formulated his famous PORC conjecture. We look at the possibilities for turning his theory into a practical algorithm for computing the number of p-class two groups of order pn for small n. We obtain the PORC formulae for the number of r-generator groups of p-class two for r  6. In addition, we obtainthe PORC formula for the number of p-class two groups of order p8.One of the ideas used in implementing Higman''s theory has led to a signi -cant speed up in Eamonn O''Brien''s ClassTwo function in Magma. In addition,we are able to simplify some of the theory. In particular, Higman''s paper con-tains ve pages of homological algebra which he uses in his proof that the number of solutions in a nite eld to a nite set of monomial equations is PORC. It turns out that the homological algebra is just razzle dazzle, and can all be replaced by the single observation that if you write the equations as the rows of a matrix then the number of solutions is the product of the elementary divisors in the Smith normal form of the matrix.