فهرست مطالب
International Journal of Group Theory
Volume:8 Issue: 1, Mar 2019
- تاریخ انتشار: 1397/10/09
- تعداد عناوین: 5
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Pages 1-9In this paper we show that every finite nonabelian p-group G in which the Frattini subgroup Φ(G) has order ≤p5 admits a noninner automorphism of order p leaving the center Z(G) elementwise fixed. As a consequence it follows that the order of a possible counterexample to the conjecture of Berkovich is at least p8Keywords: p-Groups, automorphisms, noninner automorphisms
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Pages 11-22If Gis a finite group and X a conjugacy class of elements of G, then we define \rank(G:X) to be the minimum number of elements of X generating G. In the present article, we determine the ranks for the Fischer's simple group Fi′24 and the baby monster group B.Keywords: Fischer group $Fi, {24}^{, prime}$, rank, generating triple, Baby Monster group mathbbB
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Pages 23-33For a finite group H, let cs(H) denote the set of non-trivial conjugacy class sizes of H and OC(H) be the set of the order components of H. In this paper, we show that if S is a finite simple group with the disconnected prime graph and G is a finite group such that cs(S)=cs(G), then |S|=|G/Z(G)| and OC(S)=OC(G/Z(G)). In particular, we show that for some finite simple group S, G≅S×Z(G)Keywords: Prime graph_the set of the order components of a finite group_the Schur multiplier
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Pages 35-42For a finite group G and a positive integer n, let G(n) be the set of all elements in G such that xn=1. The groups G and H are said to be of the same (order) type if |G(n)|=|H(n)|, for all n. The main aim of this paper is to show that if G is a finite group of the same type as Suzuki groups Sz(q), where q=22m+1≥8, then G is isomorphic to Sz(q) . This addresses to the well-known J. G. Thompson's problem (1987) for simple groups.Keywords: Suzuki group, Thompson's problem, Element order
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Pages 43-50A subset B of a group G is called a {\em difference basis} of G if each element g∈G can be written as the difference g=ab−1 of some elements a,b∈B. The smallest cardinality |B| of a difference basis B⊂G is called the {\em difference size} of G and is denoted by Δ[G]. The fraction ð[G]:=Δ[G]/|G|−−−√ is called the {\em difference characteristic} of G. We prove that for every n∈N the dihedral group D2n of order 2n has the difference characteristic 2–√≤ð[D2n]≤48586√≈1.983. Moreover, if n≥2⋅1015, then ð[D2n]<46√≈1.633. Also we calculate the difference sizes and characteristics of all dihedral groups of cardinality ≤80.Keywords: dihedral group, difference basis, difference characteristic