Questions about isomorphic properties using $\mathbb Z_4 , \mathbb Z_2 \times \mathbb Z_2 , P_2 $, and $V$ as examples.

Question

In Chapter 9 Exercise D1 of Pinter's "A Book of Abstract Algebra", one is asked to determine which of the following 4 groups are isomorphic to one another: $\mathbb Z_4 , \mathbb Z_2 \times \mathbb Z_2 , P_2 $, and $V$, where $V$ is the group $\{1,-1,i,-i\}$ with the operation of multiplication.

Now, having drawn up the Cayley Tables, the answers are:

$\mathbb Z_4 \cong V$

$\mathbb Z_2 \times \mathbb Z_2 \cong P_2$

In working out this problem, I noticed that each isomorphic pair had two ways of being mapped to one another. For $\mathbb Z_4 \cong V$, we have:

$0 \mapsto 1$

$2 \mapsto -1$

$1 \mapsto i$

$3 \mapsto -i$

However, I recognized that $3$ could have equivalently been mapped to $i$ with $1$ mapped to $-i$

A similar situation arises with $\mathbb Z_2 \times \mathbb Z_2 \cong P_2$, where $(0,1) \mapsto \{1\}$ and $(1,0) \mapsto \{2\}$ or, alternatively, $(0,1) \mapsto \{2\}$ and $(1,0) \mapsto \{1\}$.

So, my question is:

Is there a name for this event...where two elements, while certainly distinct, interact with other elements in an identical fashion. Further, what is the greater significance of elements exhibiting this sort of relation...do all isomorphic groups exhibit this property of "multiple mappings"?

Answer 1

Further, what is the greater significance of elements exhibiting this sort of relation...do all isomorphic groups exhibit this property of "multiple mappings"?

Essentially [see the last paragraph for the exceptions], yes.

Any two such isomorphisms $\phi_1, \phi_2 : G_1 \to G_2$ may be composed as $\phi_2^{-1} \circ \phi_1 : G_1 \to G_1$ to give an automorphism of $G_1.$ Conversely, any automorphism $\psi : G_1 \to G_1$ may be composed with one of the above isomorphisms to give another isomorphism $\phi_1 \circ \psi : G_1 \to G_2.$

In other words, what you seem to really be asking is whether the group of automorphisms $\text{Aut}(G)$ is the trivial group or not. It can be seen that if $|G| > 2,$ then $|\text{Aut}(G)| \neq 1,$ so for any pair of isomorphic groups each with more than $2$ elements, there will be more than one isomorphism between them.

On the other hand, one can also show directly that if the groups have at most two elements each, then there will only be one isomorphism between them.