On the groups satisfying the converse of Schur's theorem
Azam
Kaheni
Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran
author
Rasoul
Hatamian
Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran
author
Saeed
Kayvanfar
Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran
author
text
article
2012
eng
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.
International Journal of Group Theory
University of Isfahan
2251-7650
1
v.
4
no.
2012
1
7
http://ijgt.ui.ac.ir/article_1485_97332f3127ba88174f99b39be512bb8a.pdf
dx.doi.org/10.22108/ijgt.2012.1485
CH-groups which are finite $p$-groups
Bettina
Wilkens
Lecturer at University of Botswana
author
text
article
2012
eng
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.
International Journal of Group Theory
University of Isfahan
2251-7650
1
v.
4
no.
2012
9
23
http://ijgt.ui.ac.ir/article_1509_028ff27c245e1f41ebc5c1a1b4f12e07.pdf
dx.doi.org/10.22108/ijgt.2012.1509
Finite simple groups with number of zeros slightly greater than the number of nonlinear irreducible characters
Guangju
Zeng
the Chinese Mathematical Society
author
text
article
2012
eng
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$_{6}$.
International Journal of Group Theory
University of Isfahan
2251-7650
1
v.
4
no.
2012
25
32
http://ijgt.ui.ac.ir/article_1518_8d0839410be2400287196a5047b6c37c.pdf
dx.doi.org/10.22108/ijgt.2012.1518
Units in $\mathbb{Z}_2(C_2\times D_\infty)$
R
Sharma
Indian Institute of Technology Delhi
author
Pooja
Yadav
Kamla Nehru College,
University of Delhi, Delhi
author
Kanchan
Joshi
Department of Mathematics,
University of Delhi, Delhi
author
text
article
2012
eng
In this paper we consider the group algebra $R(C_2\times D_\infty)$. It is shown that $R(C_2\times D_\infty)$ can be represented by a $4\times 4$ block circulant matrix. It is also shown that $\mathcal{U}(\mathbb{Z}_2(C_2\times D_\infty))$ is infinitely generated.
International Journal of Group Theory
University of Isfahan
2251-7650
1
v.
4
no.
2012
33
41
http://ijgt.ui.ac.ir/article_1589_be4c70f82fd30aaa780db17dff9e42fc.pdf
dx.doi.org/10.22108/ijgt.2012.1589
Fischer matrices of Dempwolff group $2^{5}{^{\cdot}}GL(5,2)$
Ayoub
Basheer
Universities of KwaZulu-Natal and Khartoum
author
Jamshid
Moori
North-West University
author
text
article
2012
eng
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.
International Journal of Group Theory
University of Isfahan
2251-7650
1
v.
4
no.
2012
43
63
http://ijgt.ui.ac.ir/article_1590_e7bcdc949b15b34554b46d8c59cfc1ce.pdf
dx.doi.org/10.22108/ijgt.2012.1590
On Graham Higman's famous PORC paper
Michael
Vaughan-Lee
Oxford University
Mathematical Institute
author
text
article
2012
eng
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 $r\leq 6$. In addition, we obtain the PORC formula for the number of $p$-class two groups of order $p^{8}$.
International Journal of Group Theory
University of Isfahan
2251-7650
1
v.
4
no.
2012
65
79
http://ijgt.ui.ac.ir/article_1591_6206fc955e4cba86c276b42fd3c4770f.pdf
dx.doi.org/10.22108/ijgt.2012.1591