Consider the following groups: $(\mathbb{Z}_4,+)$, $(U_5,.)$, $(U_8,.)$ and the set of symmetries for a rhombus if I am not mistaken the first and last are equivalent. What other justifiable equivalencies and nonequivalencies are there and what does it mean rigorously to be in the same group in general?
Rigorous definition of what it means to be the same group?
2 Answers
Two groups $G$ and $H$ are "the same" if there is an isomorphism between them, ie a map $f: G\to H$ which is bijective ans such that for all $g,h\in G$ it satisfies $f(gh) = f(g)f(h)$ (so here the multiplication on the left is inside $G$ and on the right it is inside $H$). The reason that this is the condition we want is that in that case, anything we can say about $G$ which only uses that it is a group, we can transfer to $H$ via $f$ and vice versa.
The first two groups you mention are isomorphic, while the thirds is not isomorphic to the first two. What the symmetries of a rhombus are depends on whether you allow a square to be a rhombus. If you do not, then the group of symmetries of a rhombus is isomorphic to $U(8,\cdot)$
The term you are looking for is called a group isomorphism.
By the way if your $(U_8,\cdot)$ denotes $(\Bbb{Z}/8\Bbb{Z})^{\ast}$, it is not isomorphic to $(\Bbb{Z}_4,+)$ because one is cyclic while the other is not.