Том 15, № 3, 1963

A plexus of substitution groups was used to describe all Silov p-subgroups of an even symmetrical group. The power of a set of isomorphics proved equal to a continuum. Each of them is characterized by certain invariants — namely, by some whole p-adic number and a certain invariant which is a finite or countable cardinal number.

The analytical properties of elastic scattering amplitudes are studied in the frame of perturbation theory. Dispersion relations are derived for fixed t (or s), as well as for fixed cos 8. From these latter dispersion relations spectral representations are obtained for the partial wave amplitudes in the case of elastic scattering. The analytical properties of the nN partial wave amplitudes are also studied.

In this article which is a detailed exposition of our notes (7) and (8), we consider the differential equation (7) in which At(p) are polynomials in a complex parameter, these being uniquely determined by the condition that the equation is satisfied by the family (depending on p) of the functions (1) For non - singular values of p we build all linearly independent solutions (19) of the equation (7) and find for them integral representations (27) and (28), with the help of which we get addition theorems of type (40) and (41). If m = 2 we get the known results of the Bessel functions theory.

The authors consider the construction of Silov $p$-subgroups of a group of isometrics (linear transformations of a space preserving the metric) of a bilinear-metric space with orthogonal or simplex metrics, consisting of an orthogonal sum of two-dimensional subspaces of the form $$V_{2n} = \langle N_1, M_1\rangle \perp \langle N_2, M_2\rangle \perp ...\perp \langle N_n, M_n\rangle ,$$ where $\langle N_i, M_i \rangle $ is a linear shell of the vectors $N_i$ and $M_i$, $N_i^2 = M_i^2 = 0,\; N_i \cdot M_i = 1$ in the sense defined in the $V_{2n}$ scalar product (metrics).

The results are formulated as the following theorems.
Theorem 1. In order that subgroup $S$ of a group of isometrics of space $V_{2n}$ should be a Silov $p$-subgroup, the following conditions are necessary and sufficient:
a) subgroup $S$ must have an invariant maximal isotropic subspace $U_n \subset V_{2n}$
b) subspace $S$ must induce in $U_n$ a Silov $p$-subgroup of its full linear group.

Theorem 2. The Silov $p$-subgroup $S$ of a group of isometrics of space $V_{2n}$ is a semidirect product of the Silov $p$-subgroup of the full linear group of the vector space $U_n$ and the normal divisor $H$, which in the orthogonal metrics is isomorphic to the additive group of obliquely symmetric matrices of order n, in simplex metrics — to the additive group of symmetrical matrices of order $n$.