An Infinite Compact Group Is Necessarilyuncountable, By the Baire Category Theorem. a Compact "Hyper Group, Inwhich the Product of Two Points Is a Probability Measure, Is Much Like Acompact Group, Having an Everywhere Supported Invariant Measure, an Orthogonalsystem of Characters Which Span the Continuous Functions In the Uniformtopology, and a Multiplicative Semigroup of Positive-Definite Functions. It Isremarkable That a Compact Hyper Group Can Be Count Ably Infinite. In This Paperhyper Groups, Which Include the Algebra of Measures on the P-Adic Integerswhich Are Invariant Under the Action of the Units (For P = 2, 3, 5, * * * ) Ispresented and Investigate the Question of Whether the Spectrum or some Subsetof It Has a Hyper Group Structure.