eng
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
2015-03-01
4
1
1
6
10.22108/ijgt.2015.6212
6212
Computing character degrees via a Galois connection
Mark Lewis
lewis@math.kent.edu
1
John McVey
jmcvey@math.kent.edu
2
Department of Mathematical Sciences
Kent State University
Department of Mathematical Sciences Kent State University
In a previous paper, the second author established that, given finite fields $F < E$ and certain subgroups $C leq E^times$, there is a Galois connection between the intermediate field lattice ${L mid F leq L leq E}$ and $C$'s subgroup lattice. Based on the Galois connection, the paper then calculated the irreducible, complex character degrees of the semi-direct product $C rtimes {Gal} (E/F)$. However, the analysis when $|F|$ is a Mersenne prime is more complicated, so certain cases were omitted from that paper. The present exposition, which is a reworking of the previous article, provides a uniform analysis over all the families, including the previously undetermined ones. In the group $Crtimes{rm Gal(E/F)}$, we use the Galois connection to calculate stabilizers of linear characters, and these stabilizers determine the full character degree set. This is shown for each subgroup $Cleq E^times$ which satisfies the condition that every prime dividing $|E^times :C|$ divides $|F^times|$.
http://ijgt.ui.ac.ir/article_6212_edb9e19829eb4a1d2264f3c3f26089ed.pdf
Galois correspondence
lattice
character degree
finite field
eng
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
2015-03-01
4
1
7
12
10.22108/ijgt.2015.7277
7277
Homogenous finitary symmetric groups
Otto. H. Kegel
otto.h.kegel@math.uni-freiburg.de
1
Mahmut Kuzucuoğlu
matmah@metu.edu.tr
2
Mathematisches Institut Albert Ludwigs Universitat Eckerstr
Middle East Technical University
We characterize strictly diagonal type of embeddings of finitary symmetric groups in terms of cardinality and the characteristic. Namely, we prove the following. Let $kappa$ be an infinite cardinal. If $G=underset{i=1}{stackrel{infty}bigcup} G_i$, where $G_icong FSym(kappa n_i)$, ($H=underset{i=1}{stackrel{infty}bigcup}H_i$, where $H_icong Alt(kappa n_i)$), is a group of strictly diagonal type and $xi=(p_1, p_2, ldots )$ is an infinite sequence of primes, then $G$ is isomorphic to the homogenous finitary symmetric group $FSym(kappa)(xi)$ ($H$ is isomorphic to the homogenous alternating group $Alt(kappa)(xi))$, where $n_0=1$, $n_i=p_1p_2cdots p_i$.
http://ijgt.ui.ac.ir/article_7277_99d977c991cd0d26ba4b51775929aa07.pdf
Finitary symmetric groups
Centralizer
Locally finite simple groups
eng
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
2015-03-01
4
1
13
19
10.22108/ijgt.2015.7279
7279
On Magnus' Freiheitssatz and free polynomial algebras
Benjamin Fine
fine@fairfield.edu
1
Martin Kreuzer
martin.kreuzer@uni-passau.de
2
Gerhard Rosenberger
gerhard.rosenberger@uni-hamburg.de
3
Fairfield University
University of Passau
University of Hamburg
The Freiheitssatz of Magnus for one-relator groups is one of the cornerstones of combinatorial group theory. In this short note which is mostly expository we discuss the relationship between the Freiheitssatz and corre-sponding results in free power series rings over fields. These are related to results of Schneerson not readily available in English. This relationship uses a faithful representation of free groups due to Magnus. Using this method in free polynomial algebras provides a proof of the Freiheitssatz for one-relation monoids. We show how the classical Freiheitssatz depends on a condition on certain ideals in power series rings in noncommuting variables over fields. A proof of this result over fields would provide a completely dif erent proof of the classical Freiheitssatz.
http://ijgt.ui.ac.ir/article_7279_fcba62ba3131ea087ff25710ba67aa42.pdf
Freiheitssatz
one-relator group
Magnus representation
formal power series rings
eng
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
2015-03-01
4
1
21
32
10.22108/ijgt.2015.7376
7376
The theorems of Schur and Baer: a survey
Martyn Dixon
mdixon@ua.edu
1
Leonid Kurdachenko
lkurdachenko@i.ua
2
Aleksander Pypka
pypka@ua.fm
3
University of Alabama
Department of Algebra, Facultet of mathematic and mechanik\ National University of Dnepropetrovsk\ Gagarin prospect 72\ Dnepropetrovsk 10, 49010, Ukraine.
Department of Algebra, Facultet of mathematic and mechanik\ National University of Dnepropetrovsk\ Gagarin prospect 72\ Dnepropetrovsk 10, 49010, Ukraine.
This paper gives a short survey of some of the known results generalizing the theorem, credited to I. Schur, that if the central factor group is finite then the derived subgroup is also finite.
http://ijgt.ui.ac.ir/article_7376_0455587d5d510872dccca9f610794b9e.pdf
Schur theorem
Baer theorem
finite rank
eng
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
2015-03-01
4
1
33
39
10.22108/ijgt.2015.7326
7326
Generalizing quasinormality
John Cossey
john.cossey@anu.edu.au
1
Stewart Stonehewer
s.e.stonehewer@warwick.ac.uk
2
Australian National University
University of Warwick
Quasinormal subgroups have been studied for nearly 80 years. In finite groups, questions concerning them invariably reduce to $p$-groups, and here they have the added interest of being invariant under projectivities, unlike normal subgroups. However, it has been shown recently that certain groups, constructed by Berger and Gross in 1982, of an important universal nature with regard to the existence of core-free quasinormal subgroups generally, have remarkably few such subgroups. Therefore in order to overcome this misfortune, a generalization of the concept of quasinormality will be defined. It could be the beginning of a lengthy undertaking. But some of the initial findings are encouraging, in particular the fact that this larger class of subgroups also remains invariant under projectivities of finite $p$-groups, thus connecting group and subgroup lattice structures.
http://ijgt.ui.ac.ir/article_7326_cad8cfd369b67424cbbf096493ca294d.pdf
p-groups
quasinormal
products
eng
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
2015-03-01
4
1
41
46
10.22108/ijgt.2015.7908
7908
Groups of infinite rank with a normalizer condition on subgroups
Anna Valentina de Luca
annavalentina.deluca@unina2.it
1
Giovanna di Grazia
giovanna.digrazia@unina.it
2
Dipartimento di Matematica e Applicazioni &quot;Renato Caccioppoli&quot;- Universit&agrave; degli Studi di Napoli &quot;Federico II&quot;
Dipartimento di Matematica e applicazioni "R. Caccioppoli"-Università Federico II
Groups of infinite rank in which every subgroup is either normal or self-normalizing are characterized in terms of their subgroups of infinite rank.
http://ijgt.ui.ac.ir/article_7908_94891e4532f9aab49cc0270e018fc1c2.pdf
infinite rank
normalizer subgroup
locally graded group