International Journal of Group Theory
Volume:5 Issue: 2, Jun 2016

ýLet S S be a subset of a finite group G G ý. ýThe bi-Cayley graph BCay(G,S) BCay(G,S) of G G with respect to S S is an undirected graph with vertex set G×{1,2} G×{1,2} and edge set {{(x,1),(sx,2)}∣x∈Gý,ý s∈S} {{(x,1),(sx,2)}∣x∈Gý,ý s∈S} ý. ýA bi-Cayley graph BCay(G,S) BCay(G,S) is called a BCI-graph if for any bi-Cayley graph BCay(G,T) BCay(G,T) ý, ýwhenever BCay(G,S)≅BCay(G,T) BCay(G,S)≅BCay(G,T) we have T=gS α T=gSα for some g∈G g∈G and α∈Aut(G) α∈Aut(G) ý. ýA group G G is called a BCI-group if every bi-Cayley graph of G G is a BCI-graphý. ýIn this paperý, ýwe prove that every BCI-group is solvableý.

This work is a continuation of [Aý. ýOý. ýAsarý, ýýýOn infinitely generated groups whose proper subgroups are solvableý, ý{\em Jý. ýAlgebra}ý, ý{\bf 399} (2014) 870-886.]ý, ýwhere it was showný ýthat a perfect infinitely generated group whose proper subgroupsý are solvable and in whose homomorphic images normal closures of ýfinitely generated subgroups are residually nilpotent is a Fittingýýp p -group for a prime p p ý. ýThus this work is a study of a Fitting ýýp p -group whose proper subgroups are solvableý. ýNewý ýcharacterizations and some sufficient conditions for theý solvability of such a group are obtainedý.

In this paper we introduce the construction of free profinite products of profinite groups with commuting subgroups. We study a particular case: the proper free profinite products of profinite groups with commuting subgroups. We prove some conditions for a free profinite product of profinite groups with commuting subgroups to be proper. We derive some consequences. We also compute profinite completions of free products of (abstract) groups with commuting subgroups.

ýThe purpose of this paper is the determination of the inertiaý ýfactorsý, ýthe computations of the Fischer matrices and the ordinaryý ýcharacter table of the split extension G ¯ ¯ ¯ ¯ =ýý3 7 :Sp(6,2) G¯=ýý37:Sp(6,2) by means of Clifford-Fischer Theoryý. ýWe firstlyýýdetermine the conjugacy classes of G ¯ ¯ ¯ ¯ G¯ using the cosetý ýanalysis methodý. ýThe determination of the inertia factor groups ofý ýthis extension involved looking at some maximal subgroups of the ýmaximal subgroups of Sp 6,2). Sp(6,2). The Fischer matrices ofý ýG ¯ ¯ ¯ ¯ G¯ are all listed in this paper and their sizes rangeý ýbetween 2 and 10ý. ýThe character table of G ¯ ¯ ¯ ¯ , G¯, which is aý ý118×118 C 118×118 C -valued matrixý, ýis available in the PhDý ýthesis of the first authorý, ýwhich could be accessed onlineý.

In this paper, we examine that some simple K 4 K4 -groups can be determined uniquely by their orders and one or two irreducible complex character degrees.