From what I understand about isomorphisms is that two isomorphic groups are the same groups. They may have different names for the same elements and the operation. But the point is that the groups are same since their elements combine the same way.
So if $G \cong G'$ then every group property about $G$ also holds for $G'$, am I correct?
Now my question is— Can I interchange $G$ and $G'$ whenever and wherever I want? I thought the answer was obviously yes but… now I'm not sure.
For example: $\mathbb{Z} \cong \mathbb{2Z}$ so shouldn't $\mathbb{Z}/\mathbb{Z} = \mathbb{Z}/\mathbb{2Z}$ under the group operation $(a+H) + (b +H) = (a+b) + H$? where $H$ is either $\mathbb{Z}$ or $\mathbb{2Z}$ since both are the same groups.
But that's clearly not the case as $\mathbb{Z}/\mathbb{Z} = \{0\}$ but $\mathbb{Z}/\mathbb{2Z}=\{0,1\}$ So where does one draw the line between two isomorphic groups? How "same" are two isomorphic groups? I'm very confused.
Best Answer
This is a great question and your example with group quotients perfectly highlights why one has to be careful about notions of identity when doing maths. To answer your question,
The answer is yes, as long as you are 100% sure that you are thinking of both groups as literally just groups and nothing more (people will sometimes use the phase 'up to isomorphism' to convey this idea). This resolves the problem you had with quotients: you can't quotient an arbitrary group by another arbitrary group, the group by which you are quotienting has to be a subgroup of the other. This is extra 'data' which can be encoded in multiple ways, one nice way is to consider the special injection that embeds the one group inside the other (which doesn't exist for two arbitrary groups). So to recap, subgroups aren't just groups, they're groups with some extra data and $2\mathbb{Z}$ and $\mathbb{Z}$ are different subgroups of $\mathbb{Z}$ despite being 'the same' groups (i.e. isomorphic).