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TitleOn Recognition by Order and Degree Pattern of Finite Simple Groups
Vol.39(2) (2015) page: 163-172
Author(s): B. Akbari and A.R. Moghaddamfar  

Let GK(G) be the prime graph associated with a finite group G and D(G) be the degree pattern of G. A finite group G is said to be k-fold OD-characterizable if there exist exactly k non-isomorphic groups H such that |H|=|G| and D(H)=D(G). A 1-fold OD-characterizable group is simply called OD-characterizable. The purpose of this paper is twofold. First, it lists a number of such simple groups that have been already investigated. Second, it shows that the simple groups L6(3) and U4(5) are OD-characterizable, too.

Keywords: Prime graph; Degree pattern; Simple group.
Size:160KB   download
TitleSome Properties of Auto-Coset Groups
Vol.39(2) (2015) page: 173-179
Author(s): M. Badrkhani Asl, M.R.R. Moghaddam and M.J. Sadeghifard  

We call G to be an auto-coset group, if it contains a proper characteristic subgroup N such that xN={α(x) | α ∈ Aut(G)}, for all x ∈ G\N. Clearly, if N is a normal subgroup and α runs over the inner automorphisms of G, one obtains the notion of con-cos groups, which was first introduced by Muktibodh in 2006. In the present article we discuss some properties of the new notion. We also introduce the concept of n-auto-coset and n-auto-Camina groups and give their connections to the groups with the property P, where P is the class of all finite groups such that their absolute centre is the same as autocommutator subgroup of order p.


Keywords: Con-cos group; Auto-extra-special p-group; Auto-Camina group.
Size:103KB   download
TitleQuasirecognition by Prime Graph of the Simple Group Bn(2)
Vol.39(2) (2015) page: 181-193
Author(s): S.S. Beynekalaei, A. Iranmanesh and M.F. Ghasemabadi  

Let G be a finite group. The prime graph GK(G) of G is defined as follows. The vertices of GK(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge if G has an element of order pq. In this paper as the main result, we show that if G is a finite group such that GK(G)=GK(Bn(2)), where n ≥ 8 is even, then G has a unique nonabelian composition factor isomorphic to Bn(2).

Keywords: Finite simple groups and their classification; Prime graph; Quasirecognition.
Size:169KB   download
TitleGroups in Which Every Normal Subgroup of Infinite Rank Has Finite Index
Vol.39(2) (2015) page: 195-201
Author(s): M. De Falco, F. De Giovanni and C. Musella  

An infinite group is called just-infinite if all its non-trivial normal subgroups have finite index. It is proved here that if all normal subgroups of infinite rank of a radical group G have finite index, then G has finite rank.

Keywords: Just-infinite; Finite index; Finite rank; Infinite rank.
Size:102KB   download
TitleSome New Ideas and Results on Partially Soluble Finite Groups
Vol.39(2) (2015) page: 203-218
Author(s): W. Guo, K.P. Shum and A.N. Skiba   

In this paper, some new ideas, recent results and some open questions in the theory of finite partially soluble groups will be given and discussed.

Keywords: Finite groups; F-hypercentre; Soluble groups; Supersoluble groups; Nilpotent groups; Quasisoluble groups; Quasisupersoluble groups; Quasinilpotent groups; Saturated formations; Baer-local formations; The generalized Fitting subgroups.
Size:168KB   download
TitleOn Auto-nipoltent Groups
Vol.39(2) (2015) page: 219-224
Author(s): S. Hoseini, M.R.R. Moghaddam and S. Tajnia   

The concept of autonilpotent groups was introduced by Parvaneh and Mogha-
ddam in 2010. In the present article, we give some classification of such groups. In final section, we introduce self-auto-permutable subgroups for finite groups and obtain some more properties of autonilpotent groups.

Keywords: Autonilpotent group; Permutable subgroup; Auto-permutable subgroup; Self-auto-permutable subgroup.
Size:97KB   download
TitleOn Q??-Supplemented Subgroups of Finite Groups
Vol.39(2) (2015) page: 225-233
Author(s): A. Mahboob, L. Zhang and Y.F. Liu   

A subgroup H of G is called QΦ-supplemented in G if there exists a subgroup T of G such that G=HT and (H ∩T)HG/HG ≤ Φ(H/HG), where HG is the maximal normal subgroup of G contained in H and Φ(H/HG) is the Frattini subgroup of H/HG. In this paper, we study the properties of the QΦ-supplemented subgroups and use them to determine the structure of finite groups.

Keywords: Sylow p-subgroup; Q??-supplemented subgroup; p-nilpotent group; Supersoluble group.
Size:145KB   download
TitleA Quantitative Characterization of Mathieu Group M12
Vol.39(2) (2015) page: 235-248
Author(s): A.R. Moghaddamfar and S. Rahbariyan   

Given a group G, we denote by πe(G) the set of orders of elements in G. For a natural number m, we denote by sm(G) the number of elements of order m in G, and put nse(G)={sm(G)  |  m ∈ πe(G)}. The group M is said to be characterizable by NSE if, for every group G, the equality nse(G)=nse(M) implies that G ≅ M. In the present article, we prove that the Mathieu group M12 is characterizable by NSE. This finishes the proof of the characterizability by NSE of all Mathieu simple groups. Thus, some recent results given by C.G. Shao and Q.H. Jiang in 2014 are strengthened in this paper.

Keywords: Mathieu group M12; Element order.
Size:172KB   download
TitleOn a Maximal Subgroup of O+10 (2)
Vol.39(2) (2015) page: 249-263
Author(s): J. Moori and T.T. Seretlo  

The group ?DG = 28:O8+(2) has order 44590694400 and is a maximal subgroup of index 527 of O10+(2). However 210 + 16.O10+(2) is a maximal subgroup of the monster M = F1. The group ?DG has three inertia factor groups namely, O8+(2), SP(6,2) and 26:A8 of indices 1, 120 and 135 respectively in O8+(2). The aim of this paper is to compute the Fischer-Clifford matrices of ?DG, which together with associated partial character tables of the inertia factor groups, are used to compute the full character table of ?DG. There are 53 Fischer-Clifford matrices with sizes between 1 ×1 and 6 ×6. We also compute the abstract specification of ?DG.

Keywords: Group extensions; Orthogonal groups; Character table; Clifford theory; Inertia groups; Fischer-Clifford matrices.
Size:194KB   download
TitleOn Auto-Engle Groups and Auto-Bell Groups
Vol.39(2) (2015) page: 265-272
Author(s): M.A. Rostamyari, A.K. Mousavi and M.R.R. Moghaddam   

The third author et al. in 2013, introduced and studied the concept of auto-Bell groups. A group G is said to be n-auto-Bell if [xn,α]=[x,αn], for all x in G, α in Aut(G) and any integer n\noindentt = 0,1. Let Ln(G) be the nth-absolute centre of the group G, then some sharp bounds for the exponents of G/L2(G) and G/L3(G) are constructed, when G is an n-auto-Bell abelian or n-auto-Bell groups. Finally, we show that ⟨ϕx,α⟩ is nilpotent of class at most 4, for every inner automorphism ϕx induced by an element x of G and α in Aut(G), when G is a 3-auto-Engel group.

Keywords: Auto-Bell group; Auto-Engel group; Auto-Kappe group; Autocommutator subgroup.
Size:116KB   download
TitleSome New Results on Auto-isoclinism of Groups
Vol.39(2) (2015) page: 273-279
Author(s): M.J. Sadeghifard, M. Eshrati and M.R.R. Moghaddam  

P. Hall in 1940 introduced the concept of isoclinism between two groups, which is weaker than isomorphism. In this article, the concept of autoisoclinism will be introduced and studied some of its properties. Among other results, we show that the properties of autonilpotency and autosolubility are invariant under this new notion.

Keywords: Isoclinism; Autoisoclinism; Autonilpotent; Autosoluble groups.
Size:92KB   download
TitleOn the Classification of 2-Step Nilpotent Zinbiel 3-Algebras
Vol.39(2) (2015) page: 279-290
Author(s): F. Saeedi and S.N. Akbarossadat   

In this paper, we introduce the concept of Zinbiel n-algebra and study some of its properties. Also, we explain the relation between Leibniz n-algebras and Zinbiel n-algebras. Finally, we classify some nilpotent Zinbiel n-algebras.

Keywords: Leibniz n-algebra; Zinbiel n-algebra; Nilpotent Zinbiel algebra.
Size:129KB   download
12 Records!


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