On the Regular Representations of Algebras

Abstract

abelian group of order p(m + 1)/2 which contains no operator of order p2 just as in the case when G involves no operator of order p2. In the present case the group may be extended arbitrarily by an operator of order p or by an operator of order p2 except that at least one of the (m -1)/2 extending operators must be of order p2. Hence there result m -2 distinct groups since the extending operators can always be so selected that they have different orders except in one case. When G involves an invariant operator of order p2, all the possible groups can be constructed by starting with the abelian subgroup of type 2, 1(m -2)/2. Since the extending operator can be selected so as to have either of two different orders in each of the (m -2)/2 possible cases, the number of the possible non-abelian G's in this case is again m -2. As there is also one possible abelian group it results that, including direct products there are m -1 groups of order pm which have the property that each of them involves invariant operators of order p2 and contains exactly m -1 independent generators, m being even. When m is odd this number is m -2.In the special case when p = 2 a necessary and sufficient condition that a group of order pm has m -1 independent generators is that the squares of all its operators generate the subgroup of order 2. It is known that the number of these groups of order 2' is 3(m -1)/2 when m is odd and (3m -4)/2 when m is even.' Although these groups are somewhat more complex than those relating to the more general case when p is odd they seem to require no consideration here since their fundamental properties were determined in the article to which we referred at the close of the preceding sentence and in those referred to therein.The regular representations play an important r6le in the work of Molien, Cartan and Frobenius' in the theory of hypercomplex numbers. More recently, the theory of groups of linear transformations has been extended and new concepts have been introduced. Our first aim was to study the regular representations of an algebra with regard to these new ideas. 236

On sets of matrices with coefficients in a division ring

“…The regular representation 37 of ® is a top constituent of wX®o, and ®0 is an end constituent of « X3t. and Nesbitt [4], Nesbitt [19], Nakayama [18]. Proof.…”

“…The following two sections contain an application of the group-theoretical methods to the study of the irreducible and the Loewy constituents of a set of matrices. In Section 6, a number of further remarks are added, for instance a generalization of a theorem of A. H. Clifford (4).…”

“…Hence §*®i = ®i, i.e., &*Q®i. We can define a homomorphic mapping of upon ^j/Ü-i®i/ § *®i = ®/®i by (4) -> H*$?Qii = H*®i, H in Since §3--i/ §j is simple, this is an isomorphism. It follows that Q * is a complete residue system $i of ® (mod ®i).…”

“…This gives (cf. (1.2A)) (4.5B) If 31 and 93, (12), are two similar sets of square matrices which break up into irreducible constituents, then it is possible to carry 93 into % by a succession of similarity transformations of types (3.3A), (3.3B), and (3.3C)(1S). Similarly, 8i(2l) is the maximal completely reducible set which can appear as a top constituent of 21; and if 21 splits into 8i(2l) and 93, then §*w(*)~8«(S).…”

On the Dimension of Modules and Algebras, I

“…In fact, denoting by A L the algebra obtained from A by passing to the algebraic closure L of K, it was proved in [4] and [5] A is a symmetric algebra and therefore also a Frobenius algebra (see [2]). …”