Let $X$ be an $H-space$, and let a multiplication $,\cdot,$ be given, associative up to homotopy.
Let $\Omega X$ be the loopspace of $X$ based at the identity and let the multiplication $ \circ $ on the loopspace be given by concatenating loops.
We also have a multiplication on the loopspace given by pointwise multiplication of loops. Since the loops are based at the identity this also give a loop based at the identity. Call this multiplication $\mu$.
Are the multiplication maps $\mu$, $\circ$ homotopic?
I am trying to verify the assertion in Odd primary exponents of Moore Spaces that if the power map $\dot k: X \to X$ sending $x$ to a choice of the $k$th power $x^k=(x(x(x(x...))$ $k$ times, is null homotopic, then the map $k^\circ$ sending $\gamma \in \Omega X$ to its kth homotopy exponent is also null homotopic.
Define $k^\mu$ to be the $k$-th power map on the loopspace using pointwise multiplication of loops.
Now if $\mu$ and $\circ$ are indeed homotopic, then since $k^\mu$ is null homotopic, it follows that $k^\circ$ is also null homotopic.
So is it true that $\mu$ and $\circ$ are homotopic?