TITLE: How "same" are two isomorphic groups?
QUESTION [25 upvotes]: 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.

REPLY [2 votes]: Topological and illustrated version of Arthur's nice answer:
A trefoil knot $K_3$ in $\Bbb R^3$ cannot be unknotted but it can be in $\Bbb R^4$. Topologically a trefoil knot $K_3$ and a circle are homeomorphic i.e. one can deform one to other without cutting or gluing. So these two (one in $\Bbb R^3$ and one in $\Bbb R^4$) are the same but one uncovers a branch of mathematics known as Knot_theory while other one is fruitless from this point of view and application.

I think theses type of "same things" is a categorical notion but I am not sure.
same is true for $y=x$ and $y=\exp(x)$; both topologically are homeomorphic to $\Bbb R$, but one contains only positive numbers. One can say similar properties by considering them as additive and multiplicative groups.
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