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1-simplex abelian group Amer assume automorphism b.i.b. design Banach algebra bounded Cartan subalgebra characteristic closed commutative compact completes the proof condition consider contains continued fraction converges Corollary corresponding decomposition defined definition denote direct sum disjoint element equation equivalent exists field finite group fixed points following theorem function graphs group G Hence homomorphism hypothesis ideal idempotents identity implies incidence structure induction integer irreducible character isomorphic lattice Lemma Let G Lie algebras linear Math matrix maximal measure metric space multiplication nilpotent non-empty non-negative non-zero normal subgroup notation obtain orthogonal p-group partial direct sum partition permutation Perron plane polynomial positive integer prime Proc proof of Theorem Proposition prove quasi-compact representation respect result ring root S-functions satisfies semigroup sequence space subgroup of G subset sufficient Suppose Sylow p-subgroup symmetric topological space topology unique valuation vector vertex zero