University of IsfahanInternational Journal of Group Theory2251-76501420121201On the groups satisfying the converse of Schur's theorem17148510.22108/ijgt.2012.1485ENAzam KaheniDepartment of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, IranRasoul HatamianDepartment of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, IranSaeed KayvanfarDepartment of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, IranJournal Article20120512A 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.http://ijgt.ui.ac.ir/article_1485_97332f3127ba88174f99b39be512bb8a.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76501420121201CH-groups which are finite $p$-groups923150910.22108/ijgt.2012.1509ENBettina WilkensLecturer at University of BotswanaJournal Article20120702In their paper "<em>Finite groups whose noncentral commuting elements have centralizers of equal size</em>", 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.http://ijgt.ui.ac.ir/article_1509_028ff27c245e1f41ebc5c1a1b4f12e07.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76501420121201Finite simple groups with number of zeros slightly greater than the number of nonlinear irreducible characters2532151810.22108/ijgt.2012.1518ENGuangju Zengthe Chinese Mathematical SocietyJournal Article20120519The 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$_{6}$.http://ijgt.ui.ac.ir/article_1518_8d0839410be2400287196a5047b6c37c.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76501420121201Units in $mathbb{Z}_2(C_2times D_infty)$3341158910.22108/ijgt.2012.1589ENR SharmaIndian Institute of Technology DelhiPooja YadavKamla Nehru College,
University of Delhi, DelhiKanchan JoshiDepartment of Mathematics,
University of Delhi, DelhiJournal Article20120423In this paper we consider the group algebra $R(C_2times D_infty)$. It is shown that $R(C_2times D_infty)$ can be represented by a $4times 4$ block circulant matrix. It is also shown that $mathcal{U}(mathbb{Z}_2(C_2times D_infty))$ is infinitely generated.http://ijgt.ui.ac.ir/article_1589_be4c70f82fd30aaa780db17dff9e42fc.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76501420121201Fischer matrices of Dempwolff group $2^{5}{^{cdot}}GL(5,2)$4363159010.22108/ijgt.2012.1590ENAyoub Basheer MohammedBasheerUniversities of KwaZulu-Natal and KhartoumJamshid MooriNorth-West UniversityJournal Article20120507In [U. Dempwolff, On extensions of elementary abelian groups of order $2^{5}$ by $GL(5,2)$, <em>Rend. Sem. Mat. Univ. Padova</em>, <strong>48</strong> (1972) 359 - 364.] 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. <br /> http://ijgt.ui.ac.ir/article_1590_e7bcdc949b15b34554b46d8c59cfc1ce.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76501420121201On Graham Higman's famous PORC paper6579159110.22108/ijgt.2012.1591ENMichael Vaughan-LeeOxford University
Mathematical InstituteJournal Article20120214We investigate Graham Higman's paper <em>Enumerating</em> $p$-<em>groups</em>, II, in which he formulated his famous PORC conjecture. We are able to simplify some of the theory. In particular, Higman's paper contains five pages of homological algebra which he uses in his proof that the number of solutions in a finite field to a finite set of <em>monomial</em> 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. We obtain the PORC formulae for the number of $r$-generator groups of $p$ -class two for $rleq 6$. In addition, we obtain the PORC formula for the number of $p$-class two groups of order $p^{8}$.http://ijgt.ui.ac.ir/article_1591_6206fc955e4cba86c276b42fd3c4770f.pdf