scholarly journals On finite groups with the Cayley invariant property

1997 ◽  
Vol 56 (2) ◽  
pp. 253-261
Author(s):  
Cai Heng Li
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Cayley Graphs ◽  
Invariant Property ◽  
Frobenius Groups ◽  
Nonabelian Simple Group ◽  
A Finite Group ◽  
Ci Property

A finite group G is said to have the m-CI property if, for any two Cayley graphs Cay(G, S) and Cay(G, T) of valency m, Cay(G, S) ≅ Cay(G, T) implies Sσ = T for some automorphism σ of G. In this paper, we investigate finite groups with the m-CI property. We first construct groups with the 3-CI property but not with the 2-CI property, and then prove that a nonabelian simple group has the 3-CI property if and only if it is A5. Finally, for infinitely many values of m, we construct Frobenius groups with the m-CI property but not with the nontrivial k-CI property for any k < m.

n-RECOGNITION BY PRIME GRAPH OF THE SIMPLE GROUP PSL(2,q)

2008 ◽  
Vol 07 (06) ◽  
pp. 735-748 ◽  
Author(s):  
BEHROOZ KHOSRAVI
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Prime Number ◽  
Finite Groups ◽  
Prime Graph ◽  
Split Extension ◽  
A Finite Group

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge if there is an element in G of order pq. It is proved that if p > 11 and p ≢ 1 (mod 12), then PSL(2,p) is uniquely determined by its prime graph. Also it is proved that if p > 7 is a prime number and Γ(G) = Γ(PSL(2,p2)), then G ≅ PSL(2,p2) or G ≅ PSL(2,p2).2, the non-split extension of PSL(2,p2) by ℤ2. In this paper as the main result we determine finite groups G such that Γ(G) = Γ(PSL(2,q)), where q = pk. As a consequence of our results we prove that if q = pk, k > 1 is odd and p is an odd prime number, then PSL(2,q) is uniquely determined by its prime graph and so these groups are characterizable by their prime graph.

scholarly journals A Study About One Generation of Finite Simple Groups and Finite Groups

2021 ◽  
Vol 13 (3) ◽  
pp. 59
Author(s):  
Nader Taffach
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Prime Numbers ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
A Finite Group

In this paper, we study the problem of how a finite group can be generated by some subgroups. In order to the finite simple groups, we show that any finite non-abelian simple group can be generated by two Sylow p1 - and p_2 -subgroups, where p_1&nbsp; and p_2&nbsp; are two different primes. We also show that for a given different prime numbers p&nbsp; and q , any finite group can be generated by a Sylow p -subgroup and a q -subgroup.

On mutually permutable products of generalized supersoluble subgroups of finite groups

2016 ◽  
Vol 09 (03) ◽  
pp. 1650054
Author(s):  
E. N. Myslovets
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Paper Properties ◽  
Supersoluble Groups ◽  
A Finite Group ◽  
Mutually Permutable Products ◽  
Composition Factors

Let [Formula: see text] be a class of finite simple groups. We say that a finite group [Formula: see text] is a [Formula: see text]-group if all composition factors of [Formula: see text] are contained in [Formula: see text]. A group [Formula: see text] is called [Formula: see text]-supersoluble if every chief [Formula: see text]-factor of [Formula: see text] is a simple group. In this paper, properties of mutually permutable products of [Formula: see text]-supersoluble finite groups are studied. Some earlier results on mutually permutable products of [Formula: see text]-supersoluble groups (SC-groups) appear as particular cases.

scholarly journals Recognition of symplectic and orthogonal groups of small dimensions by spectrum

2019 ◽  
Vol 18 (12) ◽  
pp. 1950230
Author(s):  
Mariya A. Grechkoseeva ◽  
Andrey V. Vasil’ev ◽  
Mariya A. Zvezdina
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Simple Groups ◽  
Orthogonal Groups ◽  
A Finite Group ◽  
Field Automorphism ◽  
Orders Of Elements

We refer to the set of the orders of elements of a finite group as its spectrum and say that finite groups are isospectral if their spectra coincide. In this paper, we determine all finite groups isospectral to the simple groups [Formula: see text], [Formula: see text], and [Formula: see text]. In particular, we prove that with just four exceptions, every such finite group is an extension of the initial simple group by a (possibly trivial) field automorphism.

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}.

Finite Groups whose Powers have no Countably Infinite Factor Groups

1969 ◽  
Vol 21 ◽  
pp. 965-969 ◽  
Author(s):  
M. J. Billis
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
A Finite Group ◽  
Countably Infinite ◽  
Inheritance Properties

Let P be the class of all finite groups G whose powers GI have no countably infinite factor groups. Neumann and Yamamuro (1) proved that if G is a finite non-Abelian simple group, then G ∈ P. We generalize this result by proving the following theorem.THEOREM. A finite group G ∈ P if and only if G is perfect2. Inheritance properties ofP.P1. If G ∈ P and N is normal in G, then G/N ∈ P.Proof. Since (G/N)I is isomorphic to GI/NI it is clear that factor groups of (G/N)I are isomorphic to factor groups of GI, and hence finite or uncountable.P2. If G = HK, where H ∈ P and K ∈ P, then G ∈ P.Proof. We show that homomorphic images of GI are either finite or uncountable. Let ϕ be a homomorphism of GI. Then GIϕ = (HK)Iϕ = (HI/KI)ϕ = (HIϕ) (KIϕ). Since HIϕ and KIϕ must be finite or uncountable, the conclusion follows.

On the spectrum of Cayley graphs related to the finite groups

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6419-6429 ◽  
Author(s):  
Modjtaba Ghorbani ◽  
Farzaneh Nowroozi-Larki
Keyword(s):  
Finite Group ◽  
Cayley Graph ◽  
Finite Groups ◽  
Cayley Graphs ◽  
Prime Numbers ◽  
A Finite Group

Let G be a finite group of order pqr where p > q > r > 2 are prime numbers. In this paper, we find the spectrum of Cayley graph Cay(G,S) where S ? G \ {e} is a normal symmetric generating subset.

The nilpotency index of the radicals of group algebras of finite groups whose Sylow 3-subgroups are extra-special of order 27 of exponent 3

1988 ◽  
Vol 108 (1-2) ◽  
pp. 117-132
Author(s):  
Shigeo Koshitani
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Positive Integer ◽  
Simple Group ◽  
Finite Groups ◽  
Jacobson Radical ◽  
Group Algebras ◽  
Nilpotency Index ◽  
Characteristic 3 ◽  
A Finite Group

SynopsisLet J(FG) be the Jacobson radical of the group algebra FG of a finite groupG with a Sylow 3-subgroup which is extra-special of order 27 of exponent 3 over a field F of characteristic 3, and let t(G) be the least positive integer t with J(FG)t = 0. In this paper, we prove that t(G) = 9 if G has a normal subgroup H such that (|G:H|, 3) = 1 and if H is either 3-solvable, SL(3,3) or the Tits simple group 2F4(2)'.

scholarly journals On isomorphisms of connected Cayley graphs, III

1998 ◽  
Vol 58 (1) ◽  
pp. 137-145 ◽  
Author(s):  
Cai Heng Li
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Simple Group ◽  
Prime Divisor ◽  
Cayley Graphs ◽  
Simple Groups ◽  
Linear Groups ◽  
Finite Simple Groups ◽  
Natural Question ◽  
A Finite Group

For a finite group G and a subset S of G which does not contain the identity of G, we use Cay(G, S) to denote the Cayley graph of G with respect to S. For a positive integer m, the group G is called a (connected) m-DCI-group if for any (connected) Cayley graphs Cay(G, S) and Cay(G, T) of out-valency at most m, Sσ = T for some σ ∈ Aut(G) whenever Cay(G, S) ≅ Cay(G, T). Let p(G) be the smallest prime divisor of |G|. It was previously shown that each finite group G is a connected m-DCI-group for m ≤ p(G) − 1 but this is not necessarily true for m = p(G). This leads to a natural question: which groups G are connected p(G)-DCI-groups? Here we conjecture that the answer of this question is positive for finite simple groups, that is, finite simple groups are all connected 2-DCI-groups. We verify this conjecture for the linear groups PSL(2, q). Then we prove that a nonabelian simple group G is a 2-DCI-group if and only if G = A5.

scholarly journals Non-Abelian finite groups whose character sums are invariant but are not Cayley isomorphism

2019 ◽  
Vol 18 (01) ◽  
pp. 1950013
Author(s):  
Alireza Abdollahi ◽  
Maysam Zallaghi
Keyword(s):  
Finite Group ◽  
Cayley Graph ◽  
Finite Groups ◽  
Closed Subset ◽  
Cayley Graphs ◽  
Character Sums ◽  
Main Question ◽  
Vertex Set ◽  
A Finite Group ◽  
Positive Integers

Let [Formula: see text] be a group and [Formula: see text] an inverse closed subset of [Formula: see text]. By a Cayley graph [Formula: see text], we mean the graph whose vertex set is the set of elements of [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text]. A group [Formula: see text] is called a CI-group if [Formula: see text] for some inverse closed subsets [Formula: see text] and [Formula: see text] of [Formula: see text], then [Formula: see text] for some automorphism [Formula: see text] of [Formula: see text]. A finite group [Formula: see text] is called a BI-group if [Formula: see text] for some inverse closed subsets [Formula: see text] and [Formula: see text] of [Formula: see text], then [Formula: see text] for all positive integers [Formula: see text], where [Formula: see text] denotes the set [Formula: see text]. It was asked by László Babai [Spectra of Cayley graphs, J. Combin. Theory Ser. B 27 (1979) 180–189] if every finite group is a BI-group; various examples of finite non-BI-groups are presented in [A. Abdollahi and M. Zallaghi, Character sums of Cayley graph, Comm. Algebra 43(12) (2015) 5159–5167]. It is noted in the latter paper that every finite CI-group is a BI-group and all abelian finite groups are BI-groups. However, it is known that there are finite abelian non-CI-groups. Existence of a finite non-abelian BI-group which is not a CI-group is the main question which we study here. We find two non-abelian BI-groups of orders 20 and 42 which are not CI-groups. We also list all BI-groups of orders up to 30.

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