Smallest group for finite groups. Does it make sense to talk about a smallest group imaginable in terms of this group being unique? I know that given the prerequisites for a group (closure, identity, inverse, associativity) and if we don't necessarily require that the subgroup be a generator for the set I could construct a group for example: $\langle \mathbb{Z}_n, +\rangle$ for the smallest possible n, but would there not be other groups containing only 1 element? Also if we are dealing with non-finite groups what would be the possible smallest group imaginable?
(Sorry beforehand if this is poorly formulated as I have only recently started to learn about group theory.)
 A: Of course there's a group with a single element $\{e\}$ and it is unique up to isomorphism. The group operation is given by $(e,e)\mapsto e$, no other choice by the way. It is also known as the trivial group. And every group contains it as a subgroup.
Except for the trivial group every finite group has a minimal subgroup. Simply because finite groups have finitely many subgroups, and nontrival groups have at least $2$ subgroups.
Infinite groups on the other hand need not have minimal subgroups, e.g. $\mathbb{Z}$ has an infinite chain of descending groups, never ending, never stabilizng, i.e.
$$\mathbb{Z}\supseteq 2\mathbb{Z}\supseteq 4\mathbb{Z}\supseteq\cdots\supseteq 2^n\mathbb{Z}\supseteq\cdots$$
A: Note that for induction purposes (in case of finite groups) and classification, one is interested in deriving structure and properties of so-called minimal non-$\mathfrak{X}$ groups, where $\mathfrak{X}$ is a property of groups. This means that all proper subgroups of $G$ possess the property $\mathfrak{X}$, but $G$ itself not. For $\mathfrak{X}$ being abelian, Miller and Moreno classified these groups already in 1903, while later on, O.J. Schmidt did the same for $\mathfrak{X}$ being nilpotent. Minimal non-$M$ groups (monomial groups, a class between nilpotent and solvable) were treated late 1970ties by R.W. van der Waall (see for example here).
