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1.
Manoj K. Yadav 《Proceedings Mathematical Sciences》2008,118(1):1-11
We give a sufficient condition on a finite p-group G of nilpotency class 2 so that Aut
c
(G) = Inn(G), where Aut
c
(G) and Inn(G) denote the group of all class preserving automorphisms and inner automorphisms of G respectively. Next we prove that if G and H are two isoclinic finite groups (in the sense of P. Hall), then Aut
c
(G) ≃ Aut
c
(H). Finally we study class preserving automorphisms of groups of order p
5, p an odd prime and prove that Aut
c
(G) = Inn(G) for all the groups G of order p
5 except two isoclinism families. 相似文献
2.
Manoj K. Yadav 《代数通讯》2013,41(12):4576-4592
We obtain certain results on a finite p-group whose central automorphisms are all class preserving. In particular, we prove that if G is a finite p-group whose central automorphisms are all class preserving, then d(G) is even, where d(G) denotes the number of elements in any minimal generating set for G. As an application of these results, we obtain some results regarding finite p-groups whose automorphisms are all class preserving. 相似文献
3.
Zahedeh Azhdari 《代数通讯》2013,41(10):4133-4139
Let G be a group and Autc(G) be the group of all central automorphisms of G. We know that in a finite p-group G, Autc(G) = Inn(G) if and only if Z(G) = G′ and Z(G) is cyclic. But we shown that we cannot extend this result for infinite groups. In fact, there exist finitely generated nilpotent groups of class 2 in which G′ =Z(G) is infinite cyclic and Inn(G) < C* = Autc(G). In this article, we characterize all finitely generated groups G for which the equality Autc(G) = Inn(G) holds. 相似文献
4.
Let G be a group, let M and N be two normal subgroups of G. We denote by Aut N M (G), the set of all automorphisms of G which centralize G/M and N. In this paper we investigate the structure of a group G in which one of the Inn(G) = Aut N M (G), Aut N M (G) ≤ Inn(G) or Inn(G) ≤ Aut N M (G) holds. We also discuss the problem: “what conditions on G is sufficient to ensure that G has a non-inner automorphism which centralizes G/M and N”. 相似文献
5.
Mehdi Shabani Attar 《Archiv der Mathematik》2007,89(4):296-297
Let G be a group and let Aut
c
(G) be the group of central automorphisms of G. Let
be the set of all central automorphisms of G fixing Z(G) elementwise. In this paper we prove that if G is a finite p-group, then
= Inn(G) if and only if G is abelian or G is nilpotent of class 2 and Z(G) is cyclic.
This work was supported in part by the Center of Excellence for Mathematics, University of Isfahan, Iran.
Received: 30 October 2006 相似文献
6.
An automorphism α of a group G is called a commuting automorphism if each element x in G commutes with its image α(x) under α. Let A(G) denote the set of all commuting automorphisms of G. Rai [Proc. Japan Acad., Ser. A 91 (5), 57–60 (2015)] has given some sufficient conditions on a finite p-group G such that A(G) is a subgroup of Aut(G) and, as a consequence, has proved that, in a finite p-group G of co-class 2, where p is an odd prime, A(G) is a subgroup of Aut(G). We give here very elementary and short proofs of main results of Rai. 相似文献
7.
We classify completely three-generator finite p-groups G such that Ф(G)≤Z(G)and|G′|≤p2.This paper is a part of the classification of finite p-groups with a minimal non-abelian subgroup of index p,and solve partly a problem proposed by Berkovich. 相似文献
8.
Let G be a group, Aut(G) and L(G) denote the full automorphisms group and absolute centre of G, respectively. The automorphism \({\alpha\in Aut(G)}\) is called autocentral if \({g^{-1}\alpha(g)\in L(G)}\), for all \({g\in G}\). In the present paper, we investigate the properties of such automorphisms. 相似文献
9.
P. K. Chaurasia 《Analysis Mathematica》2009,35(1):15-35
Let G denote a locally compact abelian group and H a separable Hilbert space. Let L p (G, H), 1 ≤ p < ∞, be the space of H-valued measurable functions which are in the usual L p space. Motivated by the work of Helgason [1], Figa-Talamanca [11] and Bachelis [2, 3], we have defined the derived space of the Banach space L p (G, H) and have studied its properties. Similar to the scalar case, we prove that if G is a noncompact, locally compact abelian group, then L p 0 (G, H) = {0} holds for 1 ≤ p < 2. Let G be a compact abelian group and Γ be its dual group. Let S p (G, H) be the L 1(G) Banach module of functions in L p (G, H) having unconditionally convergent Fourier series in L p -norm. We show that S p (G, H) coincides with the derived space L p 0 (G, H), as in the scalar valued case. We also show that if G is compact and abelian, then L p 0 (G, H) = L 2(G, H) holds for 1 ≤ p ≤ 2. Thus, if F ∈ L p (G, H), 1 ≤ p < 2 and F has an unconditionally convergent Fourier series in L p -norm, then F ∈ L 2(G, H). Let Ω be the set of all functions on Γ taking only the values 1, ?1 and Ω* be the set of all complex-valued functions on Γ having absolute value 1. As an application of the derived space L p 0 (G, H), we prove the following main result of this paper. Let G be a compact abelian group and F be an H-valued function on the dual group Γ such that $$ \sum \omega (\gamma )F(\gamma )\gamma $$ is a Fourier-Stieltjes series of some measure µ ∈ M(G, H) for every scalar function ω such that |ω(γ)| = 1. Then F ∈ l 2(Γ, H). 相似文献
10.
A finite group G is called n-decomposable if every proper non-trivial normal subgroup of G is a union of n distinct conjugacy classes of G. In some research papers, the question of finding all positive integer n such that there is an n-decomposable finite group was posed. In this paper, we investigate the structure of 9- and 10-decomposable non-perfect finite groups. We prove that a non-perfect group G is 9-decomposable if and only if G is isomorphic to Aut(PSL(2,32)), Aut(PSL(3,3)), the semi-direct product Z 3 ∝(Z 5×Z 5) or a non-abelian group of order pq, where p and q are primes and p?1=8q, and also, a non-perfect finite group G is 10-decomposable if and only if G is isomorphic to Aut(PSL(2,17)), PSL(2,25):23, a split extension of PSL(2,25) by Z 2 in ATLAS notation (Conway et al., Atlas of Finite Groups, [1985]), Aut(U 3(3)) or D 38, where D 38 denotes the dihedral group of order 38. 相似文献
11.
Peyman Niroomand 《Ricerche di matematica》2012,61(2):341-346
By a well-known result of Green (Proc R Soc A 237:574?C581, 1956) and the formal definition of Ellis and Wiegold (Bull Austral Math Soc 60:191?C196, 1999), there is an integer t, say corank(G), such that ${|\mathcal{M}(G)| = p^{\frac{1}{2}n(n-1)-t}}$ . In Niroomand (J Algebra 322:4479?C4482, 2009), the author showed for a non-abelian group G, corank(G)????log p (|G|)?2 and classified the structure of all non-abelian p-groups of corank log p (|G|)?2. In the present paper, we are interesting to characterize the structure of all p-groups of corank log p (|G|)?1. 相似文献
12.
Rolf Brandl 《Israel Journal of Mathematics》2013,195(1):473-479
Let G be a finite p-group, and let ν(G) denote the number of conjugacy classes of non-normal subgroups of G. It is known that either ν(G) ≤ 1 or ν(G) ≥ p. We determine all p-groups G with ν(G) ≤ p + 1. 相似文献
13.
M. Shabani Attar 《印度理论与应用数学杂志》2011,42(3):183-186
Let G be a non-abelian finite p-group such that |Z
2(G)| = p
2. In this paper we prove that each maximal subgroup M ≠ C
G
(Z
2(G)) is non-abelian and has cyclic centre. 相似文献
14.
The Automorphism Group of a Class of Nilpotent Groups with Infinite Cyclic Derived Subgroups 下载免费PDF全文
The automorphism group of a class of nilpotent groups with infinite cyclic derived subgroups is determined. Let G be the direct product of a generalized extraspecial Z-group E and a free abelian group A with rank m, where E ={(1 kα_1 kα_2 ··· kα_nα_(n+1) 0 1 0 ··· 0 α_(n+2)...............000...1 α_(2n+1)000...01|αi∈ Z, i = 1, 2,..., 2 n + 1},where k is a positive integer. Let AutG G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G of G, and AutG/ζ G,ζ GG be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζ G of G. Then(i) The extension 1→ Aut_(G') G→ AutG→ Aut(G')→ 1 is split.(ii) Aut_(G') G/Aut_(G/ζ G,ζ G)G≌Sp(2 n, Z) ×(GL(m, Z)■(Z~)m).(iii) Aut_(G/ζ G,ζ GG/Inn G)≌(Z_k)~(2n)⊕(Z)~(2nm). 相似文献
15.
Mohammad Hassan Abbaspour Houshang Behravesh 《Applied mathematics and computation》2011,218(3):658-661
By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. For a given finite group G, let p(G) denote the minimal degree of a faithful representation of G by permutation matrices, and let c(G) denote the minimal degree of a faithful representation of G by quasi-permutation matrices. See [4]. It is easy to see that c(G) is a lower bound for p(G). Behravesh [H. Behravesh, The minimal degree of a faithful quasi-permutation representation of an abelian group, Glasg. Math. J. 39 (1) (1997) 51-57] determined c(G) for every finite abelian group G and also [H. Behravesh, Quasi-permutation representations of p-groups of class 2, J. Lond. Math. Soc. (2) 55 (2) (1997) 251-260] gave the algorithm of c(G) for each finite group G. In this paper, we first improve this algorithm and then determine c(G) and p(G) for an arbitrary minimal non-abelian p-group G. 相似文献
16.
M. Chaboksavar M. Farrokhi Derakhshandeh Ghouchan F. Saeedi 《Archiv der Mathematik》2014,102(5):401-409
Let G be a group and L(G) be the absolute center of G, that is, the set of all elements of G fixed by all automorphisms of G. In this paper, we classify all finite groups G whose absolute central factors are isomorphic to a cyclic group, \({\mathbb{Z}_p \times \mathbb{Z}_p}\) , D 8, Q 8, or a non-abelian group of order pq for some distinct primes p and q. 相似文献
17.
Sumana Hatui 《Proceedings Mathematical Sciences》2018,128(4):49
Let G be a finite p-group of order \(p^n\) and M(G) be its Schur multiplier. It is a well known result by Green that \(|M(G)|= p^{\frac{1}{2}n(n-1)-t(G)}\) for some \(t(G) \ge 0\). In this article, we classify non-abelian p-groups G of order \(p^n\) for \(t(G)=\log _p(|G|)+1\). 相似文献
18.
Teresa Cortadellas 《代数通讯》2013,41(2):601-612
In this article, we first give a necessary and sufficient condition on a finite purely nonabelian p-group G for the group Aut c (G) of central automorphisms of G to be elementary Abelian. We then generalize our result to the homocyclic case. 相似文献
19.
A subgroup of index p k of a finite p-group G is called a k-maximal subgroup of G. Denote by d(G) the number of elements in a minimal generator-system of G and by δ k (G) the number of k-maximal subgroups which do not contain the Frattini subgroup of G. In this paper, the authors classify the finite p-groups with δd(G)(G) ≤ p2 and δd(G)?1(G) = 0, respectively. 相似文献
20.
B. M. Veretennikov 《Proceedings of the Steklov Institute of Mathematics》2012,278(1):139-151
We study metabelian Alperin groups, i.e., metabelian groups in which every 2-generated subgroup has a cyclic commutator subgroup. It is known that, if the minimum number d(G) of generators of a finite Alperin p-group G is n ≥ 3, then d(G′) ≤ C n 2 for p≠ 3 and d(G′) ≤ C n 2 + C n 3 for p = 3. The first section of the paper deals with finite Alperin p-groups G with p≠ 3 and d(G) = n ≥ 3 that have a homocyclic commutator subgroup of rank C n 2 . In addition, a corollary is deduced for infinite Alperin p-groups. In the second section, we prove that, if G is a finite Alperin 3-group with homocyclic commutator subgroup G- of rank C n 2 + C n 3 , then G″ is an elementary abelian group. 相似文献