I've been trying to understand the group theory proof of Fermat's little theorem.
Let's say there exists group $G = (ℤ/pℤ)^x$, and it has multiplicative subgroup $H$ (a monoid group, as i understand). The group $H$ is generated by some element $a$, this makes $H$ a cyclic group.
According to group theory, $|H|$ is the smallest positive integer $k$ such that $a^k = e$ (where $e$ is identity) and considering that identity in multiplicative group is always $1$, we can assume that order of the subgroup $|H|$ is equal to output of Carmichael's function. Thus $a^k ≡ 1 mod(p)$.
Now as i understand, $|H|≡λ(p)$, if $H$ is multiplicative group, is this always a case? If so, why?