What does "the subgroups of $G$ form a chain" mean?

Question

I am being asked to show that: $G$ is a cyclic $p$-group $\iff$ its subgroups form a chain.

What does "its subgroups form a chain" mean?

Please keep in mind that I am just asking for the meaning of that phrase.

Answer 1

The subsets of $G$ form a partially ordered set with respect to the inclusion $\subseteq$; the subgroups are a subset of this partially ordered set.

For any partiall ordered set $(S,\leq)$ a subset $C \subseteq S$ is called a chain if $C$ is totally ordered with respect to $\leq$.

So the subgroups of $G$ forming a chain means that the subgroups of $G$ are totally ordered with respect to the subgroup relation $\subseteq$.

Answer 2

It means that the subgroups of $G$ can be ordered linearly with respect to inclusion: $$1=H_1 \subseteq H_2 \subseteq \cdots \subseteq H_n = G$$