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If G is a finite linear group of degree n, that is, a finite group of automorphisms of an n‐dimensional complex vector space (or, equivalently, a finite group of non‐singular matrices of order n with complex coefficients), we shall say that G is a quasi‐permutation group if the trace of every element of G is a non‐negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n, its elements, considered as acting on the elements of a basis of an n‐dimensional complex vector space V, induce automorphisms of V forming a group isomorphic to G. The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x, and so is a non‐negative integer. Thus, a permutation group of degree n has a representation as a quasi‐permutation group of degree n. See [8].

In [2], we gave algorithms to calculate c(G), q(G) and p(G) for a finite group G. In this paper, we show that for a finite p-group G, where p is a prime, q(G)=p(G). Moreover, for odd prime p, c(G)=q(G)=p(G).

Let G be a finite linear group of degree n; that is, a finite group of automorphisms of an n-dimensional complex vector space (or, equivalently, a finite group of non-singular matrices of order n with complex coefficients). We shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n, its elements, considered as acting on the elements of a basis of an n -dimensional complex vector space V, induce automorphisms of V forming a group isomorphic to G. The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x, and so is a non-negative integer. Thus, a permutation group of degree n has a representation as a quasi-permutation group of degree n. See [5].

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