Which of the following statements are true:
1. every countable group G has only countably many distinct subgroups.
2. any automorphism of the group $\mathbb{Q}$ under addition is of the form $x→qx$ for some $q\in\mathbb{Q}$
3. all non-trivial proper subgroups of $(\mathbb{R},+)$ are cyclic.
4. every infinite abelian group has at least one element of infinite order.
5. there is an element of order $51$ in the multiplicative group $(\mathbb{Z}/103\mathbb{Z})^*$
My thoughts:
1. true as union of uncountable number of countable set is uncountable
2. true as any homomorphism must be one of those form
3. false as $(\mathbb{Q},+)$ is not cyclic.
4. false example circle group.
5. true by Fermat's little theorem.
Are my guesses correct?
Best Answer
(1) Let $G$ be the direct sum of copies of $\Bbb Z/2\Bbb Z$ indexed by $\Bbb N$. $G$ is countably infinite, but each subset $A\subseteq\Bbb N$ generates a distinct subgroup $G_A=\{x\in G:x_n=0\text{ for all }n\in\Bbb N\setminus A\}$ of $G$, and $\Bbb N$ has uncountably many subsets.
(4) The statement is indeed false; the group $G$ above is an infinite Abelian group in which every non-zero element has order $2$. However, the circle group is not a counterexample: $e^{i\theta}$ has infinite order iff $\frac{\theta}{2\pi}$ is irrational, so the circle group has uncountably many elements of infinite order and only countably many of finite order.