Vol. 24, No. 2, 1968

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Semigroup algebras that are group algebras

Donald Brooks Coleman

Vol. 24 (1968), No. 2, 247–256

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 KG?

The following theorems are proved:

1. Let S have odd order n, 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 KlSKG for a group G, then S is a group.

2. Let K be a cyclotomic field over the rationals, and let G be an abelian group. Then KGKS 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 pk-th roots of unity and the cyclic group of order pk 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 K1SKG for a group G, then S is a group.

The semigroup ring of a semilattice is also considered.

Mathematical Subject Classification
Primary: 20.90
Received: 8 November 1966
Published: 1 February 1968
Donald Brooks Coleman