A characterization of HN

In a typical finite simple group of Lie type the defining characteristic $p$ is easily recognisable from the subgroup structure, since the maximal $p$-local subgroups look completely different from the maximal $r$-local subgroups, where $r$ is any prime other than $p$. There are various ways of abstracting these properties of the $p$-local subgroups, which play an important role in both the description and the classification of the finite simple groups. Such abstract definitions of `characteristic' usually assign the alternating groups no characteristic at all, whereas some of the sporadic simple groups have two or more characteristics.\par The Harada-Norton group $HN$ seems in this way to have characteristics $2$, $3$, and $5$. The main theorem of the paper under review is that $HN$ is characterised by certain conditions on the $2$-local and $5$-local subgroups (the $3$-local subgroups are not mentioned), which roughly say that the group is of bicharacteristic $\{2,5\}$. [Robert Wilson (London)]