A CHARACTERIZATION OF ALTERNATING GROUPS BY ORDERS OF SOLVABLE SUBGROUPS

2004 ◽  
Vol 03 (04) ◽  
pp. 445-452 ◽  
Author(s):  
JIANXING BI ◽  
XIANHUA LI
Keyword(s):  
Finite Group ◽  
Alternating Group ◽  
Alternating Groups ◽  
Group A ◽  
A Finite Group

In this paper, we prove that a finite group G is isomorphic to the alternating group An with n≥5 if and only if they have the same set of the orders of solvable subgroups.

OD-Characterization of Certain Finite Groups Having Connected Prime Graphs

2010 ◽  
Vol 17 (01) ◽  
pp. 121-130 ◽  
Author(s):  
A. R. Moghaddamfar ◽  
A. R. Zokayi
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Symmetric Group ◽  
Finite Groups ◽  
Alternating Group ◽  
Degree Pattern ◽  
Prime Graphs ◽  
Group A ◽  
A Finite Group

The degree pattern of a finite group G denoted by D(G) was introduced in [5]. We say that G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups having the same order and same degree pattern as G. In the present article, we show that the alternating group A10 and the automorphism group Aut (McL) are 2-fold OD-characterizable, while the automorphism group Aut (J2) is 3-fold OD-characterizable and the symmetric group S10 is 8-fold OD-characterizable. It is worth mentioning that the prime graphs associated to these groups are connected and, in fact, among the groups with this property, they are the first groups which are investigated for OD-characterizability.

A new characterization of Suzuki’s simple groups

2017 ◽  
Vol 16 (11) ◽  
pp. 1750216 ◽  
Author(s):  
Jinshan Zhang ◽  
Changguo Shao ◽  
Zhencai Shen
Keyword(s):  
Finite Group ◽  
Character Formula ◽  
Simple Groups ◽  
Complex Character ◽  
Group A ◽  
Vanishing Elements ◽  
A Finite Group

Let [Formula: see text] be a finite group. A vanishing element of [Formula: see text] is an element [Formula: see text] such that [Formula: see text] for some irreducible complex character [Formula: see text] of [Formula: see text]. Denote by Vo[Formula: see text] the set of the orders of vanishing elements of [Formula: see text]. In this paper, we prove that if [Formula: see text] is a finite group such that Vo[Formula: see text], [Formula: see text], then [Formula: see text].

NSE characterization of some alternating groups

2014 ◽  
Vol 14 (02) ◽  
pp. 1550012
Author(s):  
Neda Ahanjideh ◽  
Bahareh Asadian
Keyword(s):  
Finite Group ◽  
Element Orders ◽  
Alternating Groups ◽  
Set Of Element Orders ◽  
A Finite Group

Let p ≥ 5 be a prime and n ∈ {p, p + 1, p + 2}. Let G be a finite group and πe(G) be the set of element orders of G. Assume that k ∈ πe(G) and mk(G) is the number of elements of order k in G. Set nse (G) = {mk(G) : k ∈ πe(G)}. In this paper, we show that if nse (An) = nse (G), p ∈ π(G) and p2 ∤ |G|, then G ≅ An. As a consequence of our result, we show that if nse (An) = nse (G) and |G| = |An|, then G ≅ An.

scholarly journals A generalized Frattini subgroup of a finite group

1989 ◽  
Vol 12 (2) ◽  
pp. 263-266
Author(s):  
Prabir Bhattacharya ◽  
N. P. Mukherjee
Keyword(s):  
Finite Group ◽  
Maximal Subgroups ◽  
Frattini Subgroup ◽  
Definition Of ◽  
A Finite Group

For a finite group G and an arbitrary prime p, letSP(G)denote the intersection of all maximal subgroups M of G such that [G:M] is both composite and not divisible by p; if no such M exists we setSP(G)= G. Some properties of G are considered involvingSP(G). In particular, we obtain a characterization of G when each M in the definition ofSP(G)is nilpotent.

scholarly journals Finite Groups with Certain Permutability Criteria

2019 ◽  
Vol 12 (2) ◽  
pp. 571-576 ◽  
Author(s):  
Rola A. Hijazi ◽  
Fatme M. Charaf
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroups ◽  
Group A ◽  
A Finite Group

Let G be a finite group. A subgroup H of G is said to be S-permutable in G if itpermutes with all Sylow subgroups of G. In this note we prove that if P, the Sylowp-subgroup of G (p > 2), has a subgroup D such that 1 <|D|<|P| and all subgroups H of P with |H| = |D| are S-permutable in G, then G′ is p-nilpotent.

scholarly journals A CHARACTERIZATION OF PSL(4,p) BY SOME CHARACTER DEGREE

2019 ◽  
pp. 679
Author(s):  
Younes Rezayi ◽  
Ali Iranmanesh
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Irreducible Character ◽  
Character Degree ◽  
A Finite Group ◽  
Irreducible Character Degree

‎Let G be a finite group and cd(G) be the set of irreducible character degree of G‎. ‎In this paper we prove that if  p is a prime number‎, ‎then the simple group PSL(4,p) is uniquely determined by its order and some its character degrees‎. 

On Conjugate-Permutable Subgroups of a Finite Group

2005 ◽  
Vol 12 (04) ◽  
pp. 669-676 ◽  
Author(s):  
Mingyao Xu ◽  
Qinhai Zhang
Keyword(s):  
Finite Group ◽  
Necessary Conditions ◽  
Permutable Subgroups ◽  
Group A ◽  
A Finite Group

Let G be a finite group. A subgroup H of G is called conjugate-permutable in G if HHg = HgH for any g ∈ G. A group G is called an ECP-group if every subgroup of G is conjugate-permutable in G. In this paper, we study the influence of conjugate-permutable subgroups on the structure of a finite group, especially on the nilpotency or supersolvability of the group, and give some sufficient or necessary conditions for a finite group to be an ECP-group.

On SS-supplemented subgroups of some sylow subgroups of finite groups

2021 ◽  
Author(s):  
Xuanli He ◽  
Qinghong Guo ◽  
Muhong Huang
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Necessary Conditions ◽  
Saturated Formation ◽  
Sylow Subgroups ◽  
Group A ◽  
Sufficient And Necessary Conditions ◽  
A Finite Group

Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is called to be [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] permutes with all Sylow subgroups of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-supplemented in [Formula: see text] if there exists a subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is [Formula: see text]-permutable in [Formula: see text]. In this paper, we investigate [Formula: see text]-nilpotency of a finite group. As applications, we give some sufficient and necessary conditions for a finite group belongs to a saturated formation.

scholarly journals A characterization of finite p-soluble groups of p-length one by commutator identities

1981 ◽  
Vol 30 (3) ◽  
pp. 257-263 ◽  
Author(s):  
Rolf Brandl
Keyword(s):  
Finite Group ◽  
Classical Result ◽  
Soluble Groups ◽  
Similar Description ◽  
A Finite Group ◽  
Engel Condition ◽  
Almost All

AbstractA classical result of M. Zorn states that a finite group is nilpotent if and only if it satisfies an Engel condition. If this is the case, it satisfies almost all Engel conditions. We shall give a similar description of the class of p-soluble groups of p-length one by a sequence of commutator identities.

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