Let $G$ be a group. Suppose that the map $G \to G$, $g \mapsto g^r$ is a group homomorphism for every $r \in \mathbb N$. Then $G$ is abelian. Is this also true homotopy-theoretically?
(In the ordinary case, this need only hold for $r=2$, because if $(gh)^2 = g^2h^2$, then canceling a $g$ and an $h$ we find that $hg = gh$, i.e. $G$ is abelian).
More precisely, let $G$ be a group-like $E_n$-space. For each $r \in \mathbb N$, we may pick $m_r\in E_n(r)$ where $E_n$ is an $E_n$ operad (it doesn't matter which model we choose, nor which $m_r$ we choose), and look at the map
$G \to G$, $g \mapsto m_r(g,\dots,g)$
Questions:
Is this map always a map of $E_{n-1}$-spaces?
Supposing the answer to (1) is yes, if this $E_{n-1}$-map can be made into an $E_n$ map, does it follow that the $E_n$ structure on $G$ lifts to an $E_{n+1}$ structure?
I'm not sure I've chosen the right precisification of this question; I'm open to other interpretations.