RATIONAL AND QUASI-PERMUTATION REPRESENTATIONS OF HOLOMORPHS OF CYCLIC p-GROUPS

For a finite group G, three of the positive integers governing its representation theory over C and over Q are p(G), q(G), c(G). Here, p(G) denotes the minimal degree of a faithful permutation representation of G. Also, c(G) and q(G) are, respectively, the minimal degrees of a faithful representa tion of G by quasi-permutation matrices over the fields C and Q. We have c(G) ≤ q(G) ≤ p(G) and, in general, either inequality may be strict. In this paper, we study the representation theory of the group G = Hol(Cn _p), which is the holomorph of a cyclic group of order p^n , p a prime. This group is metacyclic when p is odd and metabelian but not metacyclic when p = 2 and n ≥ 3. We explicitly describe the set of all isomorphism types of irreducible representations of G over the field of complex numbers C as well as the isomorphism types over the field of rational numbers Q. We compute the Wedderburn decomposition of the rational group algebra of G. Using the descriptions of the irreducible representations of G over C and over Q, we show that c(G) = q(G) = p(G) = p^n for any prime p. The proofs are often different for the case of p odd and p = 2.