For $\delta \ge 1$ and $n\ge 1$, consider the simplicial complex of graphs on $n$ vertices in which each vertex has degree at most $\delta$; we identify a given graph with its edge set and admit one loop at each vertex. This complex is of some importance in the theory of semigroup algebras. When $\delta =1$, we obtain the matching complex, for which it is known that there is 3-torsion in degree $d$ of the homology whenever $\left( n-4 \right)/3\le d\le \left( n-6 \right)/2$. This paper establishes similar bounds for $\delta \ge 2$. Specifically, there is 3-torsion in degree $d$ whenever

$$\frac{\left( 3\delta -1 \right)n-8}{6}\le d\le \frac{\delta \left( n-1 \right)-4}{2}.$$

The procedure for detecting torsion is to construct an explicit cycle $z$ that is easily seen to have the property that $3z$ is a boundary. Defining a homomorphism that sends $z$ to a non-boundary element in the chain complex of a certain matching complex, we obtain that $z$ itself is a non-boundary. In particular, the homology class of $z$ has order 3.

A module M over a commutative ring R has an almost trivial dual if there is no homomorphism from M onto a free R-module of countable infinite rank. Using a new combinatorial principle (the ℵn-Black Box), which is provable in ordinary set theory, we show that for every natural number n, there exist arbitrarily large ℵn-free R-modules with almost trivial duals, when R is a complete discrete valuation domain. A corresponding result for torsion modules is also obtained.

In this note we characterize the abelian groups G which have two different proper subgroups N and M such that the subgroup lattice L(G)=[0, M]∪ [N, G] is the union of these intervals.

Let $T$ be an abelian group and $\lambda$ an uncountable regular cardinal. The question of whether there is a $\lambda$-universal group $U$ among all torsion-free abelian groups $G$ of cardinality less than or equal to $\lambda$ satisfying $\Ext\left(G,T\right)=0$ is considered. $U$ is said to be $\lambda$-universal for $T$ if, whenever a torsion-free abelian group $G$ of cardinality at most $\lambda$ satisfies $\Ext\left(G,T\right)=0$, there is an embedding of $G$ into $U$. For large classes of abelian groups $T$ and cardinals $\lambda$, it is shown that the answer is consistently no, that is to say, there is a model of ZFC in which, for pairs T and $\lambda$, there is no universal group. In particular, for $T$ torsion, this solves a problem by Kulikov.

We identify a condition, which we refer to as cohesiveness, on a subgroup S of the socle G[p] — {x ∊ G : px = 0} of an abelian p-group G which is necessary for S to be the socle of an isotype subgroup of G. It is shown, when S is countable, that this condition is both necessary and sufficient. A further restriction, definable in terms of the coset valuation on G/S, leads to the notion of S being completely cohesive in G. When S is countable, this latter condition is both necessary and sufficient for S to serve as the socle of a balanced subgroup of G. Also noteworthy is the fact that if H and K are, respectively, balanced and isotype subgroups of G with H[p] = K[p], then K is necessarily balanced in G.

The class of primary abelian groups whose subsocles are purifiable is not yet completely characterized and it contains the class of direct sums of cyclic groups and torsion complete groups. In sharp constrast with this, the class of groups whose p2-bounded subgroups are purifiable consist only of those groups which are the direct sum of a bounded and a divisible group. Various tools are developed and a short application to the pure envelopes of cyclic subgroups is given in the last section.

A primary group G is said to be a l.i.b. group if every idempotent endomorphism of every basic subgroup of G can be extended to an endomorphism of G. We establish the following characterization: A primary group is a l.i.b. group if and only if it is the direct sum of a torsion complete group and a divisible group. The technique used consists of a close analysis of certain subgroups of Prufer-like primary groups.