scholarly journals A characterization of PGL (2,pn) by some irreducible complex character degrees

2016 ◽  
Vol 99 (113) ◽  
pp. 257-264 ◽  
Author(s):  
Somayeh Heydari ◽  
Neda Ahanjideh
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Cyclic Group ◽  
Group Algebra ◽  
Character Degrees ◽  
Complex Character ◽  
Complex Group ◽  
Complex Group Algebra ◽  
A Finite Group

For a finite group G, let cd(G) be the set of irreducible complex character degrees of G forgetting multiplicities and X1(G) be the set of all irreducible complex character degrees of G counting multiplicities. Suppose that p is a prime number. We prove that if G is a finite group such that |G| = |PGL(2,p) |, p ? cd(G) and max(cd(G)) = p+1, then G ? PGL(2,p), SL(2, p) or PSL(2,p) x A, where A is a cyclic group of order (2, p-1). Also, we show that if G is a finite group with X1(G) = X1(PGL(2,pn)), then G ? PGL(2, pn). In particular, this implies that PGL(2, pn) is uniquely determined by the structure of its complex group algebra.

Recognition of PSL(2,p) ×PSL(2,p) by its complex group algebra

2017 ◽  
Vol 16 (02) ◽  
pp. 1750036 ◽  
Author(s):  
Behrooz Khosravi ◽  
Zahra Momen ◽  
Behnam Khosravi ◽  
Bahman Khosravi
Keyword(s):  
Prime Number ◽  
Group Algebra ◽  
Character Degrees ◽  
Simple Groups ◽  
Group Algebras ◽  
Complex Group ◽  
Complex Group Algebra ◽  
Quasisimple Group ◽  
Natural Groups ◽  
Groups Of Lie Type

In [H. P. Tong-Viet, Simple classical groups of Lie type are determined by their character degrees, J. Algebra 357 (2012) 61–68] the following question arose: Question. Which groups can be uniquely determined by the structure of their complex group algebras? It is proved that every quasisimple group except covers of the alternating groups is uniquely determined up to isomorphism by the structure of [Formula: see text], the complex group algebra of [Formula: see text]. One of the next natural groups to be considered are the characteristically simple groups. In this paper, as the first step in this investigation we prove that if [Formula: see text] is an odd prime number, then [Formula: see text] is uniquely determined by the structure of its complex group algebra.

scholarly journals A new characterization of the automorphism groups of Mathieu groups

2021 ◽  
Vol 19 (1) ◽  
pp. 1245-1250
Author(s):  
Xin Liu ◽  
Guiyun Chen ◽  
Yanxiong Yan
Keyword(s):  
Finite Group ◽  
Prime Power ◽  
Automorphism Groups ◽  
Difficult Problem ◽  
Character Degrees ◽  
Complex Character ◽  
Power Graph ◽  
Mathieu Groups ◽  
A Finite Group

Abstract Let cd ( G ) {\rm{cd}}\left(G) be the set of irreducible complex character degrees of a finite group G G . ρ ( G ) \rho \left(G) denotes the set of primes dividing degrees in cd ( G ) {\rm{cd}}\left(G) . For any prime p, let p e p ( G ) = max { χ ( 1 ) p ∣ χ ∈ Irr ( G ) } {p}^{{e}_{p}\left(G)}=\max \left\{\chi {\left(1)}_{p}\hspace{0.08em}| \hspace{0.08em}\chi \in {\rm{Irr}}\left(G)\right\} and V ( G ) = { p e p ( G ) ∣ p ∈ ρ ( G ) } V\left(G)=\left\{{p}^{{e}_{p}\left(G)}\hspace{0.08em}| \hspace{0.1em}p\in \rho \left(G)\right\} . The degree prime-power graph Γ ( G ) \Gamma \left(G) of G G is a graph whose vertices set is V ( G ) V\left(G) , and two vertices x , y ∈ V ( G ) x,y\in V\left(G) are joined by an edge if and only if there exists m ∈ cd ( G ) m\in {\rm{cd}}\left(G) such that x y ∣ m xy| m . It is an interesting and difficult problem to determine the structure of a finite group by using its degree prime-power graphs. Qin proved that all Mathieu groups can be uniquely determined by their orders and degree prime-power graphs. In this article, we continue this topic and successfully characterize all the automorphism groups of Mathieu groups by using their orders and degree prime-power graphs.

Complex group algebras of the direct product of non-isomorphic Suzuki groups

2019 ◽  
Vol 19 (02) ◽  
pp. 2050036
Author(s):  
Morteza Baniasad Azad ◽  
Behrooz Khosravi
Keyword(s):  
Direct Product ◽  
Group Algebra ◽  
Irreducible Character ◽  
Prime Numbers ◽  
Product Formula ◽  
Character Degrees ◽  
Group Algebras ◽  
Complex Group ◽  
Complex Group Algebra ◽  
Suzuki Groups

In this paper, we prove that the direct product [Formula: see text], where [Formula: see text] are distinct numbers, is uniquely determined by its complex group algebra. Particularly, we show that the direct product [Formula: see text], where [Formula: see text]’s are distinct odd prime numbers, is uniquely determined by its order and three irreducible character degrees.

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].

1-Connected character-graphs of finite groups with non-bipartite complement

2020 ◽  
pp. 2150058
Author(s):  
Mahdi Ebrahimi ◽  
Maryam khatami ◽  
Zohreh Mirzaei
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Character Degrees ◽  
Complex Character ◽  
Connected Graphs ◽  
A Finite Group

For a finite group [Formula: see text], let [Formula: see text] be the character-graph which is built on the set of irreducible complex character degrees of [Formula: see text]. In this paper, we wish to determine the structure of finite groups [Formula: see text] such that [Formula: see text] is 1-connected with nonbipartite complement. Also, we classify all 1-connected graphs with nonbipartite complement that can occur as the character-graph [Formula: see text] of a finite group [Formula: see text].

scholarly journals PROJECTIVE SPECIAL LINEAR GROUPS PSL4(q) ARE DETERMINED BY THE SET OF THEIR CHARACTER DEGREES

2012 ◽  
Vol 11 (06) ◽  
pp. 1250108 ◽  
Author(s):  
HUNG NGOC NGUYEN ◽  
HUNG P. TONG-VIET ◽  
THOMAS P. WAKEFIELD
Keyword(s):  
Finite Group ◽  
Character Degrees ◽  
Linear Groups ◽  
Complex Character ◽  
Special Linear ◽  
Projective Special Linear Groups ◽  
Special Linear Groups ◽  
The Family ◽  
Group A ◽  
A Finite Group

Let G be a finite group and let cd (G) be the set of all irreducible complex character degrees of G. It was conjectured by Huppert in Illinois J. Math.44 (2000) that, for every non-abelian finite simple group H, if cd (G) = cd (H) then G ≅ H × A for some abelian group A. In this paper, we confirm the conjecture for the family of projective special linear groups PSL 4(q) with q ≥ 13.

A New Characterization of PSL(2, q) by Group Order and the Set of Vanishing Element Orders

2019 ◽  
Vol 26 (03) ◽  
pp. 459-466
Author(s):  
Changguo Shao ◽  
Qinhui Jiang
Keyword(s):  
Finite Group ◽  
Prime Power ◽  
Complex Character ◽  
Element Orders ◽  
Group Order ◽  
Vanishing Elements ◽  
A Finite Group

An element g in a finite group G is called a vanishing element if there exists some irreducible complex character χ of G such that [Formula: see text]. Denote by Vo(G) the set of orders of vanishing elements of G, and we prove that [Formula: see text] if and only if [Formula: see text] and [Formula: see text], where [Formula: see text] is a prime power.

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.

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.

CHARACTERIZATION BY PRIME GRAPH OF PGL(2, pk) WHERE p AND k > 1 ARE ODD

2010 ◽  
Vol 20 (07) ◽  
pp. 847-873 ◽  
Author(s):  
Z. AKHLAGHI ◽  
B. KHOSRAVI ◽  
M. KHATAMI
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Prime Number ◽  
Prime Graph ◽  
Composition Factor ◽  
Simple Groups ◽  
Almost Simple Groups ◽  
Prime Graphs ◽  
A Finite Group ◽  
Nonabelian Composition Factor

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, p′ are joined by an edge if there is an element in G of order pp′. In [G. Y. Chen et al., Recognition of the finite almost simple groups PGL2(q) by their spectrum, Journal of Group Theory, 10 (2007) 71–85], it is proved that PGL(2, pk), where p is an odd prime and k > 1 is an integer, is recognizable by its spectrum. It is proved that if p > 19 is a prime number which is not a Mersenne or Fermat prime and Γ(G) = Γ(PGL(2, p)), then G has a unique nonabelian composition factor which is isomorphic to PSL(2, p). In this paper as the main result, we show that if p is an odd prime and k > 1 is an odd integer, then PGL(2, pk) is uniquely determined by its prime graph and so these groups are characterizable by their prime graphs.

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