Number of the Subgroups of Any Abelian Group
G. A. Miller · Proceedings of the National Academy of Sciences · 1939
We may apply similar arguments in the case of the indecomposable con- stituents T, of the second regular representations Z of J/p.We have formulas analogous to ( 9) and (11).Comparing them with ( 9) and ( 11), we findIf A is normal and simple, we have only one ZE, I = 1, and the cor- responding q), gives the square of the p-index of A.' In the case of a semisimple algebra A, we can always replace K by an algebraic extension field such that with regard to this new ground field, the numbers q, and ri are to be replaced by 1.With regard to such a ground field, we have (U.) *-MISc) dp,,,p; Zp -Ed,a,a (14) p CKX = Z dpkdpx.(15) 1 For the properties of the regular representations and intertwining matrices cf.R.