Metabelian Groups and Pencils of Bilinear Forms

Abstract

Introduction. In a recent paper 1 it was shown that the problem of classification of metabelian groups of order pLn+,, which contain a given abelian group of order pfl as a maximal invariant abelian subgroup and have commutator subgroups of order pm is equivalent to the problem of classification of the matrices x1M+ x2M2 + --* + xkMk under projective transformations on the x's and elementary transformations on the square matrices M1, M2, * , Mk. The x's and the elements of the M's are of course numbers in a modular field as are also the coefficients of the transformations. The squareness of the matrices comes from the requirement that the commutator subgroup be of order pm. The situation may then be discussed in terms of the invariant factors of the matrix x1Ml + x2M2 + -* + XkM7 The argument of that paper still holds when the commutator subgroup is not of order pm and the matrices Mi are not square. In this case however we are deprived of the use of a well-developed theory of invariant factors. So far as I know the question of the conjugacy of two matrices of the above type under transformation on the x's and simultaneous transformations on rectangular M's has not been considered. It is our purpose to colnsider the groups which give rise to such matrices in the simple case where m = 4 and k = 2 and to use the results to obtain normal forms for the matrices. It will be convenient to interpret the matrices Mi as matrices of bilinear forms in which case the matrix above, which we shall denote hereafter as XlMl + X2M2, may be taken to represent a pencil of bilinear forms.