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1.
Let ν(G) be the number of conjugacy classes of non-normal subgroups of a finite group G. We obtain two new lower bounds for ν(G) when G is a non-abelian finite p-group and p is odd. More precisely, if |G| =p n , exp Z(G) = p e , and exp G/G′ =p f , let us define λ(G) = n ? e and κ(G) = n ? f. Then we prove that ν(G) ≥ p(λ(G) ?3) +2 and ν(G) ≥ p(κ(G) ?3) +2. The first bound improves the bound ν(G) ≥ λ(G) ?1 given by [10 La Haye , R. , Rhemtulla , A. ( 1999 ). Groups with a bounded number of conjugacy classes of non-normal subgroups . J. Algebra 214 : 4163 .[Crossref], [Web of Science ®] [Google Scholar]], and almost in every case, the second one improves the bound ν(G) ≥ p(k ? 1) +1 obtained by [6 Fernández-Alcober , G. A. , Legarreta , L. ( 2008 ). Conjugacy classes of non-normal subgroups in finite nilpotent groups . J. Group Theory 11 ( 3 ): 381397 .[Crossref] [Google Scholar]], where k is defined by the condition that |G′| =p k .  相似文献   

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3.
Jiangtao Shi 《代数通讯》2013,41(10):4248-4252
As an extension of Shi and Zhang's 2011 article [4 Shi , J. , Zhang , C. ( 2011 ). Finite groups with given quantitative non-nilpotent subgroups . Comm. Algebra 39 : 33463355 .[Taylor &; Francis Online], [Web of Science ®] [Google Scholar]], we prove that any finite group having at most 23 non-normal non-nilpotent proper subgroups is solvable except for G ? A 5 or SL(2, 5), and any finite group having at most three conjugacy classes of non-normal non-nilpotent proper subgroups is solvable except for G ? A 5 or SL(2, 5).  相似文献   

4.
Let G be a finite group and cs(G) be the set of conjugacy class sizes of G. In 1987, J. G. Thompson conjectured that, if G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying that cs(G) = cs(M), then G ? M. This conjecture has been proved for Suzuki groups in [5 Guiyun, C. (1996). On Thompson's conjecture. J. Algebra 185(1):184193.[Crossref], [Web of Science ®] [Google Scholar]]. In this article, we improve this result by proving that, if G is a finite group such that cs(G) = cs(Sz(q)), for q = 22m+1, then G ? Sz(q) × A, where A is abelian. We avoid using classification of finite simple groups in our proofs.  相似文献   

5.
A. Erfanian  R. Rezaei 《代数通讯》2013,41(12):4183-4197
The aim of this article is to give a generalization of the concept of commutativity degree of a finite group G (denoted by d(G)), to the concept of relative commutativity degree of a subgroup H of a group G (denoted by d(H, G)). We shall state some results concerning the new concept which are mostly new or improvements of known results given in Gustafson (1973 Gustafson , W. H. ( 1973 ). What is the probability that two group elements commute? Amer. Math. Monthly 80 : 10311304 .[Taylor &; Francis Online], [Web of Science ®] [Google Scholar]) and Moghaddam et al. (2005 Moghaddam , M. R. R. , Chiti , K. , Salemkar , A. R. ( 2005 ). n-Isoclinism classes and n-nilpotency degree of finite groups . Algebra Colloquium 12 ( 2 ): 225261 . [Google Scholar]). Moreover, we shall define the relative nth nilpotency degree of a subgroup of a group and give some results concerning this at the end of the article.  相似文献   

6.
Let G be a finite group. Then we denote ψ(G) the sum of element orders in G. In [1 Amiri , H. , Jafarian Amiri , S. M. , Isaacs , I. M. ( 2009 ). Sums of element orders in finite groups . Comm. Algebra 37 ( 9 ): 29782980 .[Taylor &; Francis Online], [Web of Science ®] [Google Scholar]] it is proved that if G is a non-cyclic group of order n, then ψ(G) < ψ(C n ), where C n is the cyclic group of order n. Here we generalize and improve this result, and also we give an application of this improvement.  相似文献   

7.
Fernando Fantino 《代数通讯》2013,41(10):4426-4434
We classify the conjugacy classes of p-cycles of type D in alternating groups. This finishes the open cases in [3 Andruskiewitsch , N. , Fantino , F. , Graña , M. , Vendramin , L. ( 2011 ). Finite-dimensional pointed Hopf algebras with alternating groups are trivial . Ann. Mat. Pura Appl 190 : 225245 .[Web of Science ®] [Google Scholar]]. Also we determine all the subracks of those conjugacy classes which are not of type D.  相似文献   

8.
Let G be a group. If the set 𝒜(G) = {α ∈Aut(G) | xα(x) = α(x)x, for all x ∈ G} forms a subgroup of Aut(G), then G is called 𝒜(G)-group. We show that the minimum order of a non-𝒜(G) p-group is p 5 for any prime p. We also find the smallest group order of a non-𝒜(G) group. This is related to a question introduced by Deaconescu, Silberberg, and Walls [4 Deaconescu , M. , Silberberg , Gh. , Walls , G. ( 2002 ). On commuting automorphisms of groups . Arch. Math 79 : 423429 .[Crossref] [Google Scholar]]. Moreover, we prove that for any prime p and for all integer n ≥ 5, there exists a non-𝒜(G) group of order p n .  相似文献   

9.
John Bradley 《代数通讯》2013,41(8):2588-2599
A group is 2-generated if it can be generated by two elements x and y. In this case y is called a mate for x. Brenner and Wiegold (1975a Brenner , J. L. , Wiegold , J. ( 1975a ). Two-generator groups. I . Michigan Math. J. 22 : 5364 .[Crossref], [Web of Science ®] [Google Scholar]) defined a finite group G to have spread r if for every set {x 1, x 2,…, x r } of distinct nontrivial elements of G, there exists an element y ? G such that G = 〈 x i , y〉 for all i. A group is said to have exact spread r if it has spread r but not r + 1. The exact spread of a group G is denoted by s(G). Ganief (1996 Ganief , M. S. ( 1996 ). 2-Generations of the Sporadic Simple Groups , Ph.D thesis , University of Natal . [Google Scholar]) in his Ph.D. thesis proved that if G is a sporadic simple group, then s(G) ≥ 2. In Ganief and Moori (2001 Ganief , M. S. , Moori , J. ( 2001 ). On the spread of the sporadic simple groups . Comm. Algebra 29 : 32393255 .[Taylor &; Francis Online], [Web of Science ®] [Google Scholar]) the second author and Ganief used probabilistic methods and established a reasonable lower bound for the exact spread s(G) of each sporadic simple group G. The present article deals with the search for reasonable upper bounds for the exact spread of the sporadic simple groups.  相似文献   

10.
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