We find an upper bound for the number of groups of order n up to isomorphism in the variety ${\mathfrak {S}}={\mathfrak {A}_p}{\mathfrak {A}_q}{\mathfrak {A}_r}$, where p, q and r are distinct primes. We also find a bound on the orders and on the number of conjugacy classes of subgroups that are maximal amongst the subgroups of the general linear group that are also in the variety $\mathfrak {A}_q\mathfrak {A}_r$.

Let $w$ be a group-word. For a group $G$, let $G_{w}$ denote the set of all $w$-values in $G$ and let $w(G)$ denote the verbal subgroup of $G$ corresponding to $w$. The group $G$ is an $FC(w)$-group if the set of conjugates $x^{G_{w}}$ is finite for all $x\in G$. It is known that if $w$ is a concise word, then $G$ is an $FC(w)$-group if and only if $w(G)$ is $FC$-embedded in $G$, that is, the conjugacy class $x^{w(G)}$ is finite for all $x\in G$. There are examples showing that this is no longer true if $w$ is not concise. In the present paper, for an arbitrary word $w$, we show that if $G$ is an $FC(w)$-group, then the commutator subgroup $w(G)^{\prime }$ is $FC$-embedded in $G$. We also establish the analogous result for $BFC(w)$-groups, that is, groups in which the sets $x^{G_{w}}$ are boundedly finite.

Suppose that $\widetilde{G}$ is a connected reductive group defined over a field $k$, and $\Gamma$ is a finite group acting via $k$-automorphisms of $\widetilde{G}$ satisfying a certain quasi-semisimplicity condition. Then the identity component of the group of $\Gamma$-fixed points in $\widetilde{G}$ is reductive. We axiomatize the main features of the relationship between this fixed-point group and the pair $\left( \tilde{G},\Gamma \right)$, and consider any group $G$ satisfying the axioms. If both $\widetilde{G}$ and $G$ are $k$-quasisplit, then we can consider their duals $\widetilde{{{G}^{*}}}$ and ${{G}^{*}}$. We show the existence of and give an explicit formula for a natural map from the set of semisimple stable conjugacy classes in ${{G}^{*}}\,(k)$ to the analogous set for $\widetilde{{{G}^{*}}}\,(k)$. If $k$ is finite, then our groups are automatically quasisplit, and our result specializes to give a map of semisimple conjugacy classes. Since such classes parametrize packets of irreducible representations of $G(k)$ and $\widetilde{G}\,(k)$, one obtains a mapping of such packets.

In this paper we give a lower bound with respect to block length for the trace of non-elliptic conjugacy classes of the Hecke groups. One consequence of our bound is that there are finitely many conjugacy classes of a given trace in anyHecke group. We show that another consequence of our bound is that class numbers are finite for related hyperbolic $\mathbb{Z}\left[ \text{ }\!\!\lambda\!\!\text{ } \right]$-binary quadratic forms. We give canonical class representatives and calculate class numbers for some classes of hyperbolic $\mathbb{Z}\left[ \text{ }\!\!\lambda\!\!\text{ } \right]$-binary quadratic forms.

We show that every orbital measure, ${\mu }_{x} $, on a compact exceptional Lie group or algebra has the property that for every positive integer either ${ \mu }_{x}^{k} \in {L}^{2} $ and the support of ${ \mu }_{x}^{k} $ has non-empty interior, or ${ \mu }_{x}^{k} $ is singular to Haar measure and the support of ${ \mu }_{x}^{k} $ has Haar measure zero. We also determine the index $k$ where the change occurs; it depends on properties of the set of annihilating roots of $x$. This result was previously established for the classical Lie groups and algebras. To prove this dichotomy result we combinatorially characterize the subroot systems that are kernels of certain homomorphisms.