Can we define a limit of a sequence of groups? This is just a random thought I had while procrastinating some group theory. Is there a meaningful way to talk about limits of groups?
I don't know if this has any use, but here is my thought process so far. We would need a topology on the "set" of all groups (but already this might be a problem because this is too big to be a set); the only idea I had would be an order topology, where G is smaller than H if there is an injective homomorphism from G to H.
 A: Yes, in some sense there's a way to do this, but it doesn't involve limits in the sense of topology. There are constructions in category theory called limits and colimits (which resemble in some ways but not in other ways limits in topology) and these can be used to build groups out of other groups in various ways. Instead of just considering when there exists an injective homomorphism between groups we use all homomorphisms; that's what the category $\text{Grp}$ of groups is and it has a much richer structure than an order.
Arguably the simplest nontrivial example of this, which produces a new group otherwise difficult to define, is the construction of the $p$-adic integers $\mathbb{Z}_p$, which can be written as the limit (in the categorical sense) of the diagram
$$\cdots \to \mathbb{Z}/p^3\mathbb{Z} \to \mathbb{Z}/p^2 \mathbb{Z} \to \mathbb{Z}/p\mathbb{Z} \to 1.$$
What this means formally would take some time to spell out; you can consult the links for details. Interestingly, $\mathbb{Z}_p$ can also be described as the metric completion of a suitable metric on $\mathbb{Z}$ called the $p$-adic metric.

There are some intuitive examples that don't fall under the categorical framework, though. For example, in some sense the "limit" of the finite cyclic groups $\mathbb{Z}/n\mathbb{Z}$ ought to be $\mathbb{Z}$, or perhaps the circle $S^1$, depending on whether $\mathbb{Z}/n\mathbb{Z}$ has "length" $n$ or $1$. This is suggested by the relationship between the discrete Fourier transform and Fourier series. But I'm not aware of a way to make this intuition precise.
Categorically you can write down a diagram involving the finite cyclic groups whose colimit is the group $\mu_{\infty} = \cup_n \mu_n$ of all roots of unity, which is the torsion subgroup of $S^1$. And you can write down another diagram whose limit is the group $\widehat{\mathbb{Z}}$ of profinite integers. These somewhat resemble $S^1$ and $\mathbb{Z}$ respectively. But I'm not aware of a procedure that gets you back exactly $S^1$ or $\mathbb{Z}$.
As another example, in some sense the "limit" of a sequence of copies of $S^1$ whose radii get bigger and bigger ought to be $\mathbb{R}$, and similarly the "limit" of a sequence of copies of $\mathbb{Z}$ where the spacing between each integer gets smaller and smaller also ought to be $\mathbb{R}$. Physicists think in these terms informally quite frequently, and this is also suggested by the relationship between Fourier series and the Fourier transform. But again I'm not aware of a way to make this intuition precise.
Again the categorical procedures get you sort of close-ish: you can write down a diagram involving copies of $\mathbb{Z}$ whose colimit is the rationals $\mathbb{Q}$, which is at least dense in $\mathbb{R}$. And you can write down another diagram involving copies of $S^1$ whose limit is a complicated group called a solenoid, which is the Pontryagin dual of $\mathbb{Q}$ and which can be written as a quotient $\mathbb{A}_{\mathbb{Q}}/\mathbb{Q}$ where $\mathbb{A}_{\mathbb{Q}}$ is the adele ring of $\mathbb{Q}$; see this math.SE question for a nice discussion. I don't have a clear sense of how much this group resembles $\mathbb{R}$. Really it's $\mathbb{A}_{\mathbb{Q}}$ itself that resembles $\mathbb{R}$, I guess.

There is also the notion of Gromov-Hausdorff convergence, which really does involve limits in the usual metric / topological sense, and which gets used in geometric group theory, but I don't know anything about it.
