Volume 9, Issue 3

Let S ( m | n , d ) be the Schur superalgebra whose supermodules correspond to the polynomial representations of the supergroup GL( m | n ) of degree d . In this paper we determine the representation type of these algebras (that is, we classify the ones which are semisimple, have finite, tame and wild representation type). Moreover, we prove that these algebras are in general not quasi-hereditary and have infinite global dimension.

Let A and B be finitely generated profinite groups. If G is isomorphic to A × B , then any short exact sequence 1 → A → G → B → 1 splits as a direct product.

We prove that the quasi-variety consisting of the groups with trivial fourth dimension subgroup is not finitely based. This answers a problem of B. I. Plotkin.

The authors investigate the structure of locally (soluble-by-finite) simple groups Ĝ and show that such groups are locally residually finite. Sufficient conditions are given for a group G to be embeddable in such a group Ĝ and an example of a locally (abelian-by-finite) simple group is constructed.

All finitely generated subsemigroups of virtually nilpotent groups admit finite Malcev presentations. (A Malcev presentation is a presentation of a special type for a semigroup that can be embedded in a group.) All automatic or asynchronous automatic semigroups embeddable into groups admit finite Malcev presentations. Finitely generated subsemigroups of virtually free groups are automatic. A finitely generated subsemigroup of the free product of a free group and an abelian group that fails to have a finite Malcev presentation is exhibited. Therefore the class of groups all of whose finitely generated subsemigroups admit finite Malcev presentations is properly contained in the class of coherent groups. Finitely generated subsemigroups of the free product of a free monoid and an abelian group are asynchronously automatic and therefore have finite Malcev presentations.