Rational representations and permutation representations of finite groups

QUASI-PERMUTATION REPRESENTATIONS OF METACYCLIC 2-GROUPS

By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. Thus, every permutation matrix over C is a quasipermutation matrix. For a given finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or of a faithful representation of G by permutation matrices), let q(G) denote the minimal degree of a fa...

quasi-permutation representations of metacyclic 2-groups

by a quasi-permutation matrix we mean a square matrix over the complex field c with non-negative integral trace. thus, every permutation matrix over c is a quasipermutation matrix. for a given finite group g, let p(g) denote the minimal degree of a faithful permutation representation of g (or of a faithful representation of g by permutation matrices), let q(g) denote the minimal degree of a fai...

Finding Minimal Permutation Representations of Finite Groups

A minimal permutation representation of a finite group G is a faithful G-set with the smallest possible cardinality. We study the structure of such representations and show that for most groups they may be obtained by a greedy construction. It follows that whenever the algorithm works (except when central involutions intervene) all minimal permutation representations have the same set of orbit ...

Constructing Rational Representations of Finite Groups

Permutation representations and rational irreducibility

The natural character π of a finite transitive permutation group G has the form 1G + θ where θ is a character which affords a rational representation of G. We call G a QI-group if this representation is irreducible over Q. Every 2-transitive group is a QI-group, but the latter class of groups is larger. It is shown that every QI-group is 3/2transitive and primitive, and that it is either almost...