Doubt regarding the identical nature of groups that are isomorphic.

Question

I have been told that $2$ isomorphic groups are necessarily identical from the standpoint of group theory, but I can't digest the fact, let a group $(G, *)$ be isomorphic to $(H, \#)$ by a certain map $\psi$, how can these two groups be identical if $*$ and $\#$ are two different operations?

Answer 1

They're not identical, only isomorphic. Consider $D_3\cong S_3$.

One case in which they are identical is when $H\le G$, $G$ is finite, and $H\cong G$.

This doesn't work if $G$ is infinite. Consider $k\Bbb Z=\{ka\mid a\in\Bbb Z\}$ for $k\notin\{0,\pm 1\}$, which is clearly a subgroup of $\Bbb Z$ where $k\Bbb Z\cong \Bbb Z$, but $k\Bbb Z\neq\Bbb Z$. We need them not to be identical in order to define the group $\Bbb Z/k\Bbb Z$ of order $|k|$.