We state a sufficient condition for a fusion system to be saturated. This is then used to investigate localities with kernels: that is, localities that are (in a particular way) extensions of groups by localities. As an application of these results, we define and study certain products in fusion systems and localities, thus giving a new method to construct saturated subsystems of fusion systems.

A subgroup H of a group G is pronormal in G if each of its conjugates $H^g$ in G is conjugate to it in the subgroup $\langle H,H^g\rangle $ ; a group is prohamiltonian if all of its nonabelian subgroups are pronormal. The aim of the paper is to show that a locally soluble group of (regular) cardinality in which all proper uncountable subgroups are prohamiltonian is prohamiltonian. In order to obtain this result, it is proved that the class of prohamiltonian groups is detectable from the behaviour of countable subgroups. Examples are exhibited to show that there are uncountable prohamiltonian groups that do not behave very well. Finally, it is shown that prohamiltonicity can sometimes be detected through the analysis of the finite homomorphic images of a group.

This paper concerns rigidity of the mapping class groups. It is shown that any homomorphism $\varphi\,{:}\,\mcg_g\,{\to}\,\mcg_h$ between mapping class groups of closed orientable surfaces with distinct genera $g\,{>}\,h$ is trivial if $g\,{\geq}\, 3$, and has finite cyclic image for all $g\,{\geq}\, 1$.

Some implications are drawn for more general homomorphs of these groups.

The paper deals with locally finite groups $G$ having an involution $\phi$ such that $C_G(\phi)$ is of finite rank. The following theorem gives a very detailed description of such groups.

Let $G$ be a locally finite group having an involution $\phi$ such that $C_G(\phi)$ is of finite rank. Then $G/[G,\phi]$ has finite rank. Furthermore, $[G,\phi]'$ contains a characteristic subgroup $B$ such that the following hold.

(1) $B$ is a product of finitely many subgroups normal in $[G,\phi]$ isomorphic to either PSL$(2,K)$ or SL$(2,K)$ for some infinite locally finite fields $K$ of odd characteristic.

(2) $[G,\phi]'/B$ has finite rank.

We introduce a notion of kernel systems on finite groups: roughly speaking, a kernel system on the finite group G consists in the data of a pseudo-Frobenius kernel in each maximal solvable subgroup of G, subject to certain natural conditions. In particular, each finite CA-group can be equipped with a canonical kernel system. We succeed in determining all finite groups with kernel system that also possess a Hall p′-subgroup for some prime factor p of their order; this generalizes a previous result of ours (Communications in Algebra 18(3), 1990, pp. 833-838). Remarkable is the fact that we make no a priori abelianness hypothesis on the Sylow subgroups.

Let KG be the group ring of the group G over the field K and U(KG) its unit group. When G is finite we derive conditions which imply that every noncentral subnormal subgroup of U(KG) contains a free group of rank two. We also show that residual nilpotence of U(KG) coincides with nilpotence, this being no longer true if G is infinite.

We can answer partially the following question: when is G sub-normal in U(KG)?