Semigroup algebras that are group algebras
Donald B. Coleman · Pacific Journal of Mathematics · 1968
If S is a finite semigroup, and if K is a field, under what conditions is there a group G such that the semigroup algebra KS is isomorphic to the group algebra KGΊThe following theorems are proved: 1.Let S have odd order n f and let K be either a real number field or GF(q), where q is a prime less than any prime divisor of n.If KS = KG for a group G, then S is a group.2. Let K be a cyclotomic field over the rationale, and let G be an abelian group.Then KG = KS for a semigroup S that is not a group if and only if for some prime p and some positive integer k, K contains all p k th roots of unity and the cyclic group of order p k is a direct factor of G. 3. Let S be a commutative semigroup of order n, and let K -GF(p), where p is a prime not exceeding the smallest prime dividing n.If KS = KG for a group G, then S is a group.The semigroup ring of a semilattice is also considered. 1* Preliminary remarks. The basic definitions and concepts