فهرست مطالب
International Journal of Group Theory
Volume:8 Issue: 2, Jan 2019
- تاریخ انتشار: 1398/09/10
- تعداد عناوین: 5
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Pages 1-10We study equations over completely simple semigroups and describe the coordinate semigroups of irreducible algebraic sets for such semigroups.Keywords: system of equations, coordinate semigroups, universal algebraic geometry
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Pages 11-24Given a finite group G and a subset S⊆G, the bi-Cayley graph \bcay(G,S) is the graph whose vertex set is G×{0,1} and edge set is {{(x,0),(sx,1)}:x∈G,s∈S}. A bi-Cayley graph \bcay(G,S) is called a BCI-graph if for any bi-Cayley graph \bcay(G,T), \bcay(G,S)≅\bcay(G,T) implies that T=gSα for some g∈G and α∈\aut(G). A group G is called an m-BCI-group if all bi-Cayley graphs of G of valency at most m are BCI-graphs. It was proved by Jin and Liu that, if G is a 3-BCI-group, then its Sylow 2-subgroup is cyclic, or elementary abelian, or \Q [European J. Combin. 31 (2010) 1257--1264], and that a Sylow p-subgroup, p is an odd prime, is homocyclic [Util. Math. 86 (2011) 313--320]. In this paper we show that the converse also holds in the case when G is nilpotent, and hence complete the classification of nilpotent 3 -BCI-groups.Keywords: bi-Cayley graph, BCI-group, graph isomorphism
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Pages 25-28Let π be a set of primes, and let G be a finite π-separable group. We consider the Isaacs Bπ-characters. We show that if N is a normal subgroup of G, then Bπ(G/N)=\irrG/N∩Bπ(G) .Keywords: rmBpi-characters, pi-theory, pi-separable groups
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Pages 29-40Let K be a commutative ring with identity and N the free nilpotent K-algebra on a non-empty set X. Then N is a group with respect to the circle composition. We prove that the subgroup generated by X is relatively free in a suitable class of groups, depending on the choice of K. Moreover, we get unique representations of the elements in terms of basic commutators. In particular, if K is of characteristic 0 the subgroup generated by X is freely generated by X as a nilpotent group.Keywords: groups of finite exponent, relatively free groups, circle group, free nilpotent algebra, algebra group
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Pages 41-46Let G be a finite group, and \Irr(G) be the set of complex irreducible characters of G. Let ρ(G) be the set of prime divisors of character degrees of G. The character degree graph of G, which is denoted by Δ(G), is a simple graph with vertex set ρ(G), and we join two vertices r and s by an edge if there exists a character degree of G divisible by rs. In this paper, we prove that if G is a finite group such that Δ(G)=Δ(\PSL2(q)) and |G|=|\PSL2(q)|, then G≅\PSL2(q) .Keywords: character degree graph, simple group, characterization