Prove $\mathbb Z/4k\mathbb Z^\times$ is acyclic for $k \ge 2$

Question

I tried proving it by showing that for every element $a$ the statement $a^{\phi(4k)/2} \equiv 1 \pmod{4k}$ is true by either induction, which did not get me anywhere.

Answer 1

In a cyclic group, at most one element has order $2$,

but, in $(\mathbb Z/4k\mathbb Z)^\times$ for $k\ge2$, both $2k+1$ and $2k-1$ have order $2$.

(You should check that.)