Are fundamental groups (at any basepoint) of a commutative H-space abelian?

Question

Let $X$ be a commutative simplicial monoid and let $s\in X$. Is $\pi_1(X,s)$ abelian? (obviously $s$ not necessarily in the same component of the identity element). The direct route would be to know if the product in $\pi_1(X,s)$ is induced by the multiplication in $X$, as in the case $s=1$ but I don't know how to see if this happens.

Answer 1

No. Let $X$ be a semigroup given by $(x,y)\mapsto x_0$ for some fixed $x_0\in X$. Extend it to the monoid $M=X\sqcup\{1\}$.

Since the same semigroup action is well defined and continuous on any $X$ then you can take $X=S^1\vee S^1$ and note that the fundamental group of $M$ is abelian only at the identity component (being a singleton).

Answer 2

Let $Y$ be any simplicial set, and let $X$ be the free simplicial commutative monoid on $Y$. Note that $X$ can be identified as a simplicial set with the disjoint union of symmetric powers $\coprod_nY^n/\Sigma_n$, and in particular $Y^1/\Sigma_1\cong Y$ is a union of connected components of $X$. Since $Y$ can be arbitrary, this means $X$ can have connected components of arbitrary homotopy type.