Does every infinite group have a subgroup?

Question

This question was asked in abstract algebra quiz and I am confused about it. So, I am asking for help here.

Does every infinite group have a subgroup?

I thought about $(\mathbb{Z},+)$ and $(\mathbb{Q}, +)$ but they have subgroup and the former implies that every infinite cyclic group have subgroup.

So, No progress could be made on this particular question.

Any help please!!

Answer 1

There always is a proper non-trivial subgroup in any infinite group.

To see why, first note that $\mathbb Z$ has a proper non-trivial subgroup, $2\mathbb Z$. So any group isomorphic to $\mathbb Z$ has a non-trivial proper subgroup.

Let $G$ be an infinite group not isomorphic to $\mathbb Z$.

Note that for any $g\neq0\in G$, there exists a group homomorphism $\mathbb Z\to G$ given by $z\mapsto g^z$. Hence, $(g)$ is isomorphic to a quotient group of $\mathbb Z$, and $(g)\neq(0)$, which means that $(g)$ is proper and non-trivial.