University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
1
4
2012
12
01
On the groups satisfying the converse of Schur's theorem
1
7
EN
Azam
Kaheni
Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran
azam.kaheni@stu-mail.um.ac.ir
Rasoul
Hatamian
Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran
hatamianr@yahoo.com
Saeed
Kayvanfar
Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran
skayvanf@yahoo.com
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.
Capable group,n-isoclinism,Extra special p-group,Schur's theorem
http://ijgt.ui.ac.ir/article_1485.html
http://ijgt.ui.ac.ir/article_1485_97332f3127ba88174f99b39be512bb8a.pdf
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
1
4
2012
12
01
CH-groups which are finite $p$-groups
9
23
EN
Bettina
Wilkens
Lecturer at University of Botswana
wilkensb@mopipi.ub.bw
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.
Finite-$p$-groups,AC-groups,conjugate rank
http://ijgt.ui.ac.ir/article_1509.html
http://ijgt.ui.ac.ir/article_1509_028ff27c245e1f41ebc5c1a1b4f12e07.pdf
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
1
4
2012
12
01
Finite simple groups with number of zeros slightly greater than the number of nonlinear irreducible characters
25
32
EN
Guangju
Zeng
the Chinese Mathematical Society
weiwei@suse.edu.cn
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$_{6}$.
finite groups,characters,zeros of characters
http://ijgt.ui.ac.ir/article_1518.html
http://ijgt.ui.ac.ir/article_1518_8d0839410be2400287196a5047b6c37c.pdf
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
1
4
2012
12
01
Units in $mathbb{Z}_2(C_2times D_infty)$
33
41
EN
R
Sharma
Indian Institute of Technology Delhi
rksharma@maths.iitd.ac.in
Pooja
Yadav
Kamla Nehru College,
University of Delhi, Delhi
iitd.pooja@gmail.com
Kanchan
Joshi
Department of Mathematics,
University of Delhi, Delhi
kanchan.joshi@gmail.com
In 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.
Unit Group,infinite dihedral group,Circulant Matrices
http://ijgt.ui.ac.ir/article_1589.html
http://ijgt.ui.ac.ir/article_1589_be4c70f82fd30aaa780db17dff9e42fc.pdf
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
1
4
2012
12
01
Fischer matrices of Dempwolff group $2^{5}{^{cdot}}GL(5,2)$
43
63
EN
Ayoub
Basheer Mohammed
Basheer
Universities of KwaZulu-Natal and Khartoum
ayoubbasheer@gmail.com
Jamshid
Moori
North-West University
jamshid.moori@nwu.ac.za
In [U. Dempwolff, On extensions of elementary abelian groups of order $2^{5}$ by $GL(5,2)$, Rend. Sem. Mat. Univ. Padova, 48 (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.
Group extensions,Dempwolff group,character table,Clifford Theory,inertia groups,Fischer matrices,Schur multiplier,projective characters,covering group
http://ijgt.ui.ac.ir/article_1590.html
http://ijgt.ui.ac.ir/article_1590_e7bcdc949b15b34554b46d8c59cfc1ce.pdf
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
1
4
2012
12
01
On Graham Higman's famous PORC paper
65
79
EN
Michael
Vaughan-Lee
Oxford University
Mathematical Institute
michael.vaughan-lee@chch.ox.ac.uk
We investigate Graham Higman's paper Enumerating $p$-groups, 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 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. 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 $p^{8}$.
Enumerating p-groups,PORC conjecture,Graham Higman
http://ijgt.ui.ac.ir/article_1591.html
http://ijgt.ui.ac.ir/article_1591_6206fc955e4cba86c276b42fd3c4770f.pdf