Let R be a ring with identity of characteristic two and G a nontrivial torsion group. We show that if the units in the group ring $RG$ are all trivial, then G must be cyclic of order two or three. We also consider the case where R is a commutative ring with identity of odd prime characteristic and G is a nontrivial locally finite group. We show that in this case, if the units in $RG$ are all trivial, then G must be cyclic of order two. These results improve on a result of Herman et al. [‘Trivial units for group rings with G-adapted coefficient rings’, Canad. Math. Bull.48(1) (2005), 80–89].

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.