Research Projects

Jhon Thompson proved that if a finite group G has a fixed point free automorphism of prime order then G is nilpotent. Neumann proved that a uniquely 2-divisible group with a fixed point free automorphism of order 2 is abelian. Araujo Jao and Kinyon Michael extended the results of Thompson and Neumann to finite inverse semigroups. They proved that if a finite inverse semigroup S has an automorphism which fixes the set of idempotents of S, then S is a nilpotent Clifford semigroup, and if a uniquely 2-divisible inverse semigroup S has an automorphism of order 2 which fixes idempotents of S then maps each element to its inverse and hence S is commutative. In this project we propose to extend these results to some other classes of semigroups which are close to groups. A property that was useful in the proof of Theorem of Araujo and Kinyon is that an automorphism of an inverse semigroup preserves the inversion map. This same property also holds for completely regular semigroups, and so it is natural to prove the result for completely regular semigroups. Cancellative semigroups (a semigroup is cancellative if it satisfies both left and right cancellation laws in the usual sense) form another natural class of semigroups closely related to groups. Therefore, the next problem is very natural. Problem : Does the analog of Theorem of Araujo and Kinyon holds for cancellative semigroups. Nilpotency in groups have several equivalent formulations. Motivated by this Kowol and Mitsch while generalizing it to inverse semigroups with central idempotents proved similar characterizations. So we also pose the following problem. Problem : Find appropriate notions of nilpotence for other classes of semigroups containing the class of all semigroups, such that the restriction to groups is equivalent to the usual one, and, in addition, a generalization of Theorem of Araujo and Kinyon holds for that class of semigroups.