scholarly journals A simple group of order 168 is isomorphic to PGL(2,7)

2020 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Simple Group ◽  
Linear Group ◽  
General Linear Group ◽  
General Linear ◽  
Projective General Linear Group

We present a sketch on a problem related to the isomorphism between the simple group of order 168 and the projective general linear group.

RECOGNITION OF THE PROJECTIVE GENERAL LINEAR GROUP PGL(2, q) BY ITS NONCOMMUTING GRAPH

2011 ◽  
Vol 10 (02) ◽  
pp. 201-218 ◽  
Author(s):  
LIANGCAI ZHANG ◽  
WUJIE SHI
Keyword(s):  
Simple Group ◽  
Linear Group ◽  
General Linear Group ◽  
Simple Groups ◽  
General Linear ◽  
Almost Simple Group ◽  
Almost Simple Groups ◽  
Vertex Set ◽  
Noncommuting Graph ◽  
Projective General Linear Group

The noncommuting graph ∇(G) of a non-abelian group G is defined as follows. The vertex set of ∇(G) is G\Z(G) where Z(G) denotes the center of G and two vertices x and y are adjacent if and only if xy ≠ yx. It has been conjectured that if P is a finite non-abelian simple group and G is a group such that ∇(P) ≅ ∇(G), then G ≅ P. In the present paper, our aim is to consider this conjecture in the case of finite almost simple groups. In fact, we characterize the projective general linear group PGL (2, q) (q is a prime power), which is also an almost simple group, by its noncommuting graph.

scholarly journals BOUNDEDNESS PROPERTIES OF AUTOMORPHISM GROUPS OF FORMS OF FLAG VARIETIES

2020 ◽  
Vol 25 (4) ◽  
pp. 1161-1184
Author(s):  
A. GULD
Keyword(s):  
Automorphism Group ◽  
Linear Group ◽  
General Linear Group ◽  
Automorphism Groups ◽  
Finite Subgroup ◽  
Flag Variety ◽  
General Linear ◽  
Roots Of Unity ◽  
Flag Varieties ◽  
Projective General Linear Group

Abstract We call a flag variety admissible if its automorphism group is the projective general linear group. (This holds in most cases.) Let K be a field of characteristic 0, containing all roots of unity. Let the K-variety X be a form of an admissible flag variety. We prove that X is either ruled, or the automorphism group of X is bounded, meaning that there exists a constant C ∈ ℕ such that if G is a finite subgroup of AutK(X), then the cardinality of G is smaller than C.

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