Exercise
The multiplicative monoid of invertible elements of Z/8Z is isomorphic to
(Z/2Z,+,0)×(Z/2Z,+,0)
(Z/2Z,+,0)×(Z/3Z,+,0)
(Z/4Z,+,0)
Yes, it has 4 elements and each element squares to e.
No, the size of the monoid we are looking for equals
the number of elements relatively prime to 8, which is 4.
No, they have the same number of elements, but the structures are different:
the invertible elements in Z/8Z are 1, 3, 5, 7, all of which square to 1.
But there are elements in
(Z/4Z,+,0) that do not "square" to 1 (but to 2).
(Since the latter monoid is written additively, squaring means multiplying by 2.)