Journal•ISSN: 1433-5883
Journal of Group Theory
De Gruyter
About: Journal of Group Theory is an academic journal published by De Gruyter. The journal publishes majorly in the area(s): Group (mathematics) & Abelian group. It has an ISSN identifier of 1433-5883. Over the lifetime, 1272 publications have been published receiving 12134 citations. The journal is also known as: Journal of group theory (Print).
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TL;DR: In this article, it was shown that a δ-hyperbolic group for δ < 1 2 is a free product F ∗ G1 ∗... ∗Gn, where G is a finite group of finite rank and each Gi is a group.
Abstract: We prove that a δ-hyperbolic group for δ < 1 2 is a free product F ∗ G1 ∗ . . . ∗Gn where F is a free group of finite rank and each Gi is a finite group.
1,284 citations
TL;DR: In this article, it was shown that the undirected power graph determines the directed power graph up to isomorphism, and that two finite groups which have isomorphic undirectED power graphs have the same number of elements of each order.
Abstract: Abstract The directed power graph of a group G is the digraph with vertex set G, having an arc from y to x whenever x is a power of y; the undirected power graph has an edge joining x and y whenever one is a power of the other. We show that, for a finite group, the undirected power graph determines the directed power graph up to isomorphism. As a consequence, two finite groups which have isomorphic undirected power graphs have the same number of elements of each order.
137 citations
TL;DR: In this article, it was shown that any finitely generated Coxeter group acts properly discontinuously on a locally finite, finite dimensional CAT(0) cube complex, and for any word hyperbolic or right angled Coxeter groups, the cubing is cocompact.
Abstract: We show that any finitely generated Coxeter group acts properly discontinuously on a locally finite, finite dimensional CAT(0) cube complex. For any word hyperbolic or right angled Coxeter group we prove that the cubing is cocompact. We show how the local structure of the cubing is related to the partial order studied by Brink and Howlett in their proof of automaticity for Coxeter groups.
131 citations
TL;DR: In particular, this article showed that all elements in SOnðqÞ are real for n 1 0 ðmod 4Þ and for n odd for GOnðkÞ with n arbitrary characters.
Abstract: According to the Berman–Witt theorem, the number of real classes of G is equal to the number of complex irreducible characters whose values are real (such characters are called real ). Each real irreducible character is the character of a real or quaternion representation of G. For this reason Problem 1.1 has attracted considerable attention for various classes of groups; see [9], [12], [13]. In particular, Feit and Zuckerman [9] studied this problem for classical groups extended by a graph automorphism. They also showed that all elements are real in the groups Sp2nðqÞ with q1 1 ðmod 4Þ, and Gow [12] proved this for q even. There are some other results in the literature which are not concerned with quasi-simple finite groups. In particular, Gow [13] proved that all elements in SOnðqÞ are real for n1 0 ðmod 4Þ and for n odd. This is also true for GOnðqÞ with n arbitrary; see [8], [13], [20]. The main result of the paper is the following theorem which completely solves Problem 1.1 for finite quasi-simple groups:
101 citations
TL;DR: For all values of q > 3, the degrees of the irreducible complex characters of every group H such that S 6 H 6 Aut(S) as discussed by the authors can be obtained.
Abstract: Denote by S the projective special linear group PSL2(q) over the field of q elements. We determine, for all values of q > 3, the degrees of the irreducible complex characters of every group H such that S 6 H 6 Aut(S). We also determine the character degrees of certain extensions of the special linear group SL2(q). Explicit knowledge of the character tables of SL2(q), GL2(q), PSL2(q), and PGL2(q) is used along with standard Clifford theory to obtain the degrees.
91 citations