TITLE: Abstract algebra T/F questions.
QUESTION [7 upvotes]: This is from our review and my study group is wondering if we can get some feedback on our progress:

$1$. The symmetric group $S_3$ only has two proper normal subgroups.
True, because $e \subset A_3 \subset S_3$.
$2$. Every abelian group is cyclic.
False, every cyclic group is abelian but the converse is not true OR a counter example is a Klein-4 group.
$3$. The ideal generated by $x^2+1$ in $\mathbb{R}[x]$ is maximal.
True. Every ideal generated by irreducible element in principal domain is maximal.
$4$. The order of the cycle $(a_1, \cdots, a_k) \in S_n$ is $k$.
False. The order should be $k-1$.
$5$. Let $f: R \to S$ be a ring homomorphism, and $I \subset S$ a maximal ideal. Then $f^{-1}(I)$ is maximal.
True. Because isomorphism preserves ring properties (honestly no idea how to do this one).

REPLY [2 votes]: Three for five isn't bad: but you need to also show there are at most two normal subgroups in $S_3$.
$(4)$ Note that the order of a cycle in $S_n$ is the length of its cycle, which in your case, is $k$. (More generally, the order of a permutation expressed as $k$ disjoint cycles is the least common multiple of the lengths of the disjoint cycles.)
$(5)$ And the inverse image of a maximal ideal is not necessarily itself maximal. As @Tobias just posted, consider the embedding of $\mathbb Z$ into $\mathbb Q$. (Note also that the question was about a ring homomorphism, not an isomorphism, which you mention in your reasoning that follows.)