In a discussion regarding Mochizuki's work on the ABC conjecture which can be found here, Peter Scholze stated that
"the datum of a group is strictly more than the datum of a group up to isomorphism"
Explicitly, in what way is this true? Can someone give an example of a group satisfying the statement and explain precisely how one loses information up to isomorphism?
Here is an example from some old research of mine. I was (habitually) identifying groups up to isomorphism, and it made me miss a very obvious fact. The idea, out of context, is this:
If we think of $\mathbb{Z}^n$ as the group of integer lattice points in $\mathbb{R}^n$, then a sublattice $\Lambda$ will actually be isomorphic to $\mathbb{Z}^n$ as a group, but will represent something entirely different!
Here is a conrete example:
Consider $\mathbb{Z}^2 \leq \mathbb{R}^2$, shown in black. Then $\mathbb{Z}^2 \cong \Lambda \leq \mathbb{Z}^2$ is a sublattice (shown as red circles) carries different geometric information, even though the groups are (abstractly) the same.
Long story short, me identifying the two different embeddings of $\mathbb{Z}^2$ in $\mathbb{R}^2$ led to a headache that was quickly resolved when I finally realized that there's more to a group than its isomorphism class.
I hope this helps ^_^