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.

Integral Cayley Graphs over Finite Groups

2020 ◽  
Vol 27 (01) ◽  
pp. 131-136
Author(s):  
Elena V. Konstantinova ◽  
Daria Lytkina
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Cyclic Group ◽  
Finite Groups ◽  
Cayley Graphs ◽  
Alternating Group ◽  
Generating Set ◽  
A Finite Group

We prove that the spectrum of a Cayley graph over a finite group with a normal generating set S containing with every its element s all generators of the cyclic group 〈s〉 is integral. In particular, a Cayley graph of a 2-group generated by a normal set of involutions is integral. We prove that a Cayley graph over the symmetric group of degree n no less than 2 generated by all transpositions is integral. We find the spectrum of a Cayley graph over the alternating group of degree n no less than 4 with a generating set of 3-cycles of the form (k i j) with fixed k, as {−n+1, 1−n+1, 22 −n+1, …, (n−1)2 −n+1}.

More on the OD-Characterizability of a Finite Group

2011 ◽  
Vol 18 (04) ◽  
pp. 663-674 ◽  
Author(s):  
A. R. Moghaddamfar ◽  
S. Rahbariyan
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Symmetric Groups ◽  
Alternating Groups ◽  
Degree Pattern ◽  
Prime Graphs ◽  
A Finite Group ◽  
P 53 ◽  
Number Of Authors

The degree pattern of a finite group G was introduced in [10]. We say that G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups with the same order and same degree pattern as G. When a group G is 1-fold OD-characterizable, we simply call it OD-characterizable. In recent years, a number of authors attempt to characterize finite groups by their order and degree pattern. In this article, we first show that for the primes p=53, 61, 67, 73, 79, 83, 89, 97, the alternating groups Ap+3 are OD-characterizable, while the symmetric groups Sp+3 are 3-fold OD-characterizable. Next, we show that the automorphism groups Aut (O7(3)) and Aut (S6(3)) are 6-fold OD-characterizable. It is worth mentioning that the prime graphs associated with all these groups are connected.

OD-Characterization of Symmetric Group S57

2013 ◽  
Vol 834-836 ◽  
pp. 1799-1802
Author(s):  
Mei Yang
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Simple Groups ◽  
Almost Simple Groups ◽  
Degree Pattern ◽  
A Finite Group

In this paper, we show that the symmetric group can be characterized by its order and degree pattern. In fact, we get the following theorem: Let G be a finite group such that and . Then G is isomorphisic to one of the almost simple groups: and . Particularly, is 3-fold OD-characterizable.

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 the Symmetric Group S49

2012 ◽  
Vol 535-537 ◽  
pp. 2596-2599 ◽  
Author(s):  
Yan Xiong Yan
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Prime Graph ◽  
Degree Pattern ◽  
A Finite Group

The degree pattern of a finite group G associated with its prime graph has been introduced in [1] and denoted by D(G). The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions |G|=|H| and D(G)=D(H).Moreover, a 1-fold OD-characterizable group is simply called an OD-characterizable group. In this paper, we will show that the symmetric group S49 can be characterized by its order and degree pattern. In fact, the symmetric group S49 is 3-fold OD-characterizable

scholarly journals Finite groups with only small automorphism orbits

2020 ◽  
Vol 23 (4) ◽  
pp. 659-696
Author(s):  
Alexander Bors
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Symmetric Group ◽  
Finite Groups ◽  
Maximum Length ◽  
Natural Action ◽  
A Finite Group

AbstractWe study finite groups G such that the maximum length of an orbit of the natural action of the automorphism group {\mathrm{Aut}(G)} on G is bounded from above by a constant. Our main results are the following: Firstly, a finite group G only admits {\mathrm{Aut}(G)}-orbits of length at most 3 if and only if G is cyclic of one of the orders 1, 2, 3, 4 or 6, or G is the Klein four group or the symmetric group of degree 3. Secondly, there are infinitely many finite (2-)groups G such that the maximum length of an {\mathrm{Aut}(G)}-orbit on G is 8. Thirdly, the order of a d-generated finite group G such that G only admits {\mathrm{Aut}(G)}-orbits of length at most c is explicitly bounded from above in terms of c and d. Fourthly, a finite group G such that all {\mathrm{Aut}(G)}-orbits on G are of length at most 23 is solvable.

scholarly journals Finite groups which admit a fixed-point-free automorphism group isomorphic to S3

1990 ◽  
Vol 48 (3) ◽  
pp. 384-396
Author(s):  
David Parrott
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Automorphism Group ◽  
Symmetric Group ◽  
Finite Groups ◽  
Even Order ◽  
A Finite Group

AbstractLet G be a finite group of even order coprime to 3. If G admits a fixed-point-free automorphism group isomorphic to the symmetric group on three letters, then we prove that G is soluble.

scholarly journals Automorphism orbits of finite groups

1986 ◽  
Vol 40 (2) ◽  
pp. 253-260 ◽  
Author(s):  
Thomas J. Laffey ◽  
Desmond MacHale
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Groups ◽  
Prime Power ◽  
Structure Theorem ◽  
Equivalence Classes ◽  
Prime Power Order ◽  
A Finite Group

AbstractLet G be a finite group and let Aut(G) be its automorphism group. Then G is called a k-orbit group if G has k orbits (equivalence classes) under the action of Aut(G). (For g, hG, we have g ~ h if ga = h for some Aut(G).) It is shown that if G is a k-orbit group, then kGp + 1, where p is the least prime dividing the order of G. The 3-orbit groups which are not of prime-power order are classified. It is shown that A5 is the only insoluble 4-orbit group, and a structure theorem is proved about soluble 4-orbit groups.

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.

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.

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