For a finite group $G$, define $l(G)=(\prod _{g\in G}o(g))^{1/|G|}/|G|$, where $o(g)$ denotes the order of $g\in G$. We prove that if $l(G)>l(A_{5}),l(G)>l(A_{4}),l(G)>l(S_{3}),l(G)>l(Q_{8})$ or $l(G)>l(C_{2}\times C_{2})$, then $G$ is solvable, supersolvable, nilpotent, abelian or cyclic, respectively.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$ be a finite group of order $n$, and let $\text {C}_n$ be the cyclic group of order $n$. For $g\in G$, let ${\mathrm{o}}(g)$ denote the order of $g$. Let $\phi $ denote the Euler totient function. We show that $\sum _{g \in \text {C}_n} \phi ({\mathrm{o}}(g))\geq \sum _{g \in G} \phi ({\mathrm{o}}(g))$, with equality if and only if $G$ is isomorphic to $\text {C}_n$. As an application, we show that among all finite groups of a given order, the cyclic group of that order has the maximum number of bidirectional edges in its directed power graph.

Let $L(V)$ be the free Lie algebra on a finite-dimensional vector space $V$ over a field $K$, with homogeneous components $L^n(V)$ for $n \geq 1$. If $G$ is a group and $V$ is a $KG$-module, the action of $G$ extends naturally to $L(V)$, and the $L^n(V)$ become finite-dimensional $KG$-modules, called the Lie powers of $V$. In the decomposition problem, the aim is to identify the isomorphism types of indecomposable $KG$-modules, with their multiplicities, in unrefinable direct decompositions of the Lie powers. This paper is concerned with the case where $G$ has prime order $p$, and $K$ has characteristic $p$. As is well known, there are $p$ indecomposables, denoted here by $J_1,\dots,J_p$, where $J_r$ has dimension $r$. A theory is developed which provides information about the overall module structure of $L(V)$ and gives a recursive method for finding the multiplicities of $J_1,\dots,J_p$ in the Lie powers $L^n(V)$. For example, the theory yields decompositions of $L(V)$ as a direct sum of modules isomorphic either to $J_1$ or to an infinite sum of the form $J_r \oplus J_{p-1} \oplus J_{p-1} \oplus \ldots $ with $r \geq 2$. Closed formulae are obtained for the multiplicities of $J_1,\dots,J_p$ in $L^n(J_p)$ and $L^n(J_{p-1})$. For $r < p-1$, the indecomposables which occur with non-zero multiplicity in $L^n(J_r)$ are identified for all sufficiently large $n$.

2000 Mathematical Subject Classification: 17B01, 20C20.