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2011•Journal of North University of ChinaRequires access

A Criterion for the Existence of Fixed Points of Groups Acting on Sets

LI Xiu-ping

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Abstract

The existence problem of fixed-points of groups on sets was studied,the notion of group action triple was introduced,and the main result was proved: Let(X,M,G) be a group action triple,where M is a nontrivial p-group for some prime p,G is p-solvable and acts irreducibly on M,if CG(M) is non-transitive on X,then G has necessarily a fixed-point on X;further,if CG(M)G,then the fixed-point is unique.This result presents a new technique for the study of related problems in the theory of groups,and an application to the character theory of finite groups was given.

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The existence problem of fixed-points of groups on sets was studied,the notion of group action triple was introduced,and the main result was proved: Let(X,M,G) be a group action triple,where M is a nontrivial p-group for some prime p,G is p-solvable and acts irreducibly on M,if CG(M) is non-transitive on X,then G has necessarily a fixed-point on X;further,if CG(M)G,then the fixed-point is unique.This result presents a new technique for the study of related problems in the theory of groups,and an application to the character theory of finite groups was given.

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Available abstract

The existence problem of fixed-points of groups on sets was studied,the notion of group action triple was introduced,and the main result was proved: Let(X,M,G) be a group action triple,where M is a nontrivial p-group for some prime p,G is p-solvable and acts irreducibly on M,if CG(M) is non-transitive on X,then G has necessarily a fixed-point on X;further,if CG(M)G,then the fixed-point is unique.This result presents a new technique for the study of related problems in the theory of groups,and an application to the character theory of finite groups was given.

Key concepts: Fixed point, Mathematics, Group (periodic table), Prime (order theory), Combinatorics, Group action, Transitive relation, Character (mathematics)

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