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Characterization of the simple components of the group algebras over the $p$ -adic number field

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Sur la décomposition des algèbres de groupes

The following theorem is proved for any prime p: Every irreducible rational represention of a finite p-group is the difference of two transitive permutation representations Also given are two useful results about representations of p-groups, which are known to experts in the field 1 Statement of results Presented here are three results about characters of finite p-groups The first two are probably known to people familiar with the subject The first follows directly from a theorem of Solomon [7, p 156, Theorem 4] and the second from a generalization of this in Feit [2, p 73, 143] Since they do not seem to have been stated and proved explicitly our purpose in doing so here is to make them available to a wider audience The third result generalizes a theorem of Graeme Segal [6] that may not be well known among group theorists As Segal states in his paper, Feit has observed that this theorem may be proved using the result above We present such a proof here and obtain the following result THEOREM An irreducible rational representation of a finite p-group, p a prime, is the difference of two transitive permutation representations Let Q denote the rational field and QG the rational group algebra of a finite group G Let p be a prime number Gal denotes a Galois group [X2 : Xi] denotes the index of X1 in X2 for either fields or groups Absolute value bars denote order All groups are finite here THEOREM 1 Let G be a p-group and X an irreducible complex character of G Then one of the following holds: (i) There exists a linear character A on a subgroup H of G which induces X and generates the same field as X; that is AG = X and Q(A) = Q(X) (ii) p = 2 and there exist subgroups H < K in G with [K : H] = 2 and a linear character A of H such that with AK, [Q(A) Q(f)] = 2, 5G = X, and Q(M) = Q(X) THEOREM 2 Let G be a p-group and X an irreducible complex character of G for which the Schur index mQ(X) = 2 Then G contains a generalized quaternion section More specifically there is an irreducible character f on a subgroup K of G Received by the editors August 20, 1986 Presented at the San Antonio Meeting of the AMS on January 22, 1987 1980 Mathematics Subject Classification (1985 Revision) Primary 20C15

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Characters of $p$-groups

Über die berechnung der schurschen indizes von Charakteren endlicher Gruppen

Schur indices in finite groups

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Louis Solomon

Papers

The representation of finite groups in algebraic number fields