Let G be a cyclic group of prime order p and K a field of characteristic p. The set of classes of isomorphic indecomposable (K, G)-modules forms a basis over the complex field for an algebra p (Green, 1962) with addition and multiplication being derived from direct sum and tensor product operations.

Algebras n with similar properties can be defined for all n ≥ 2. Each such algebra is isomorphic to a matrix algebra Mn of n × n matrices with complex entries and standard operations. The characters of elements of n are the eigenvalues of the corresponding matrices in Mn.

A Latin square is considered to be a set of n2 cells with three block systems. An automorphisni is a permutation of the cells which preserves each block system. The automorphism group of a Latin Square necessarily has at least 4 orbits on unordered pairs of cells if n < 2. It is shown that there are exactly 4 orbits if and only if the square is the composition table of an elementary abelian 2-group or the cclic group of order 3.

Non-completely simple bisimple semigroups S which are generated by a finite number of idempotents are studied by means of Rees matrix semigroups over local submonoids eSe, e = e2 ∈ S. If under the natural partial order on the set Es of idempotents of such a semigroup S the sets ω(e) = {ƒ ∈ Es: ƒ ≤ e} for each e ∈ Es are well-ordered, then S is shown to contain a subsemigroup isomorphic to Sp4, the fundamental four-spiral semigroup. A non-completely simple hisimple semigroup is constructed which is generated by 5 idempotents but which does not contain a subsemigroup isomorphic to Sp4.

Given a universal algebra A, one can define for each positive integer n the set of functions on A which can be “interpolated” at any n elements of A by a polynomial function on A. These sets form a chain with respect to inclusion. It is known for several varieties that many of these sets coincide for all algebras A in the variety. We show here that, in contrast with these results, the coincident sets in the chain can to a large extent be specified arbitrarily by suitably choosing A from the variety of commutative semigroups.

The purpose of this note is to determine the automorphism group of the doubly regular tournament of Szekeres type, and to use it to show that the corresponding skew Hadamard matrix H of order 2(q + 1), where q ≡5(mod 8) and q > 5, is not equivalent to the skew Hadamard matrix H(2q + 1) of quadratic residue type when 2q + 1 is a prime power.

Congruences on regular semigroups have been characterized in terms of normal equivalences on sets of idempotents and kernels of congruences. A revised characterization is presented here with considerably simplified expressions for the least and greatest congruences associated with normal equivalences and with a new description of kernels. The results are then applied to characterize congruences on completely regular semigroups.

If G is the special orthogonal group O+(V) of a quadratic space V over a finite field of characteristic p, and r is a positive integer, we determine the abelian p-subgroups of largest order in G whose fixed subspaces in V have dimension at least r. In particular, we determine the abelian subgroups of largest order in a Sylow p-subgroup of G, extending some results obtained with different methods by Barry (1979).

We show that the classical interpretation of H3(G, A) is equivalent to Taylor's solutions of compound extensions of groups. It is also equivalent to the exactness to an eight term sequence. Only halves of the equivalences are fully shown in the paper but the other halves are clear.

The Fitting class (of finite, soluble, groups), , is said to be Hall π-closed (where π is a set of primes) if whenever G is a group in and H is a Hall π-subgroup of G, then H belongs to . In this paper, we study the Hall π-closure of products of Fitting classes. Our main result is a characterisation of the Hall π-closedFitting classes of the form (where denotes the so-called smallest normal Fitting class), subject to a restriction connecting π with the characteristic of . We also characterise those Fitting classes (respectively, ) such that (respectively, ) is Hall π-closed for all Fitting classes . In each case, part of the proof uses a concrete group construction. As a bonus, one of these construction also yields a “cancellation result” for certain products of Fitting classes.

Let S be a regular semigroup and D(S) its associated category as defined in Loganathan (1981). We introduce in this paper the notion of an extension of a D(S)-module A by S and show that the set Ext(S, A) of equivalence classes of extensions of A by S forms an abelian group under a Baer sum. We also study the functorial properties of Ext(S, A).

In this paper we characterize the structure of finite inverse perfect semigroups and study congruences on those semigroups, in particular we study those semigroups that have modular lattice of congruences.

Every irreducible ordinary character in a p-block of a finite metabelian group is of height 0 if and only if the defect group of the p-block is abelian.

We show that every inverse semigroup is an idempotent separating homomorphic image of a convex inverse subsemigroup of a P-semigroup P(G, L, L), where G acts transitively on L. This division theorem for inverse semigroups can be applied to obtain a division theorem for pseudo-inverse semigroups.

Let T be a totally ordered set, PT the semigroup of partial transformations on T, and A(T) the l-group of order-preserving permutations of T. We show that PT is a regular left l-semigroup. Let be the set of α ∈ PT such that α is order-preserving and the domain of α is a final segment of T. Then is an l-semigroup, and we prove that it is the largest transitive l-subsemigroup of PT which contains A(T). When T is Dedekind complete, we characterize the largest regular l-semigroup of . When A(T) is also 0 − 2 transitive we show that there can be no l-subsemigroup of properly containing A(T) which is either inverse or a union of groups.

A finite variety is a class of finite groups closed under taking subgroups, factor groups and finite direct products. To each such class there exists a sequence w1, w2,… of words such that the finite group G belongs to the class if and only if wk(G) = 1 for almost all k. As an illustration of the theory we shall present sequences of words for the finite variety of groups whose Sylow p-subgroups have class c for c = 1 and c = 2.

The main results are as follows. A finitely generated soluble group G is polycyclic if and only if every infinite set of elements of G contains a pair generating a polycyclic subgroup; and the same result with “polycyclic” replaced by “coherent”.

The two problem, both raised in the literature, are: (I) Is there, amongst all the permutational products (p.p.s.) on the amalgam = (A, B; H) at least one which is a minimal generalized regular product? (II) If one of the p.p.s. on is isomorphic to the generalized free product (g.f.p.) F on U are they all? We answer both of them negatively.

A partially ordered semigroup S is said to be a Dubreil-Jacotin semigroup if there is an isotone homomorphism θ of S onto a partially ordered group such that {} has a greatest member. In this paper we investigate the structure of regular Dubreil-Jacotin semigroups in which the imposed partial order extends the natural partial order on the idempotents. The main tool used is a local structure theorem which is introduced in Section 2. This local structure theorem applies to many other contexts as well.

In 1957 P. Hall conjectured that every (finitely based) variety has the property that, for every group G, if the marginal factor-group is finite, then the verbal subgroup is also finite. The content of this paper is to present a precise bound for the order of the verbal subgroup of a G when the marginal factor-group is of order Pn (p a prime and n > 1) with respect to the variety of polynilpotent groups of a given class row. We also construct an example to show that the bound is attained and furthermore, we obtain a bound for the order of the Baer-invariant of a finite p-group with respect to the variety of polynilpotent groups.

A group G is called semi-n-abelian, if for every g ∈ G there exists at least one a(g) ∈ G-which depends only on g-such that (gh)n = a-1(g)gnhna(g) for all h ∈ G; a group G is called n-abelian, if a(g) = e for all g ∈ G. According to Durbin the following holds for n-abelian groups: If G is n-abelian for at lesast 3 consecutive integers, then G in n-abelian for all integers and these groups are exactly the abelian groups. In this paper this problem is generalized to the semi-n-abelian case: If a finite group G is semi-n-abelian for at least 4 consecutive integers then G is semi-n-abelian for all integers and these groups are exactly the nilpotent groups, where the Sylow-2-subgroup is abelian, the Sylow-3-subgroup is any element of the Levi-variety ([[g, h], h] = e ∀ g, h ∈ G) and the Sylow-p-subgroup (p < 3) is of class <2. As a consequence we get a description of all finite (3-)groups, which are elements of the Levi-variety.