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ON GROUPS FOR WHICH THE BERNSTEIN CHARACTERIZATION OF A GAUSSIAN DISTRIBUTION IS VALID1

We prove that the following properties of a locally compact Abelian group X are equivalent: (i) for any compact Corwin subgroup K of X the factor-group X=K contains no subgroup topologically isomorphic to 2-dimensional torus T 2 ; (ii) the connected component of zero of X contains no more than one element of order two. These properties of a group X are necessary and suucient in order that any Gaussian distribution in Bernstein sense on X can be represented as a convolution of a Gaussian and an idempotent distribution.