فهرست مطالب

International Journal of Group Theory - Volume:8 Issue:2, 2019
  • Volume:8 Issue:2, 2019
  • تاریخ انتشار: 1398/09/10
  • تعداد عناوین: 5
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  • Artem N. Shevlyakov * Pages 1-10

    We 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
  • Hiroki Koike, Istvan Kovacs * Pages 11-24

    ‎‎Given 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
  • Mark L. Lewis * Pages 25-28

    ‎Let 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
  • Juliane Hansmann * Pages 29-40

    ‎Let 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
  • Zeinab Akhlaghi, Maryam Khatami *, Behrooz Khosravi Pages 41-46

    ‎‎Let 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