Quotient groups and quotient rings

The Klein four-group $V = \{e, (1\,2)(3\,4), (1\,3)(2\,4), (1\,4)(2\,3)\}$ is normal in $S_4$. What is the order of the quotient group $S_4/V$?

Let $H = \{0, 4, 8\}$, a subgroup of $\Z/12\Z$. What is the order of the element $5 + H$ in the quotient group $(\Z/12\Z)/H$?

To show that $(aN)(bN) = abN$ is well defined, suppose $a^{-1}a' \in N$ and $b^{-1}b' \in N$ and write $(ab)^{-1}(a'b') = \big(b^{-1}(a^{-1}a')b\big)\big(b^{-1}b'\big)$. At which point is the normality of $N$ needed?

Which of these quotient rings are fields? (Select all that apply.)

Which homomorphism, combined with the first isomorphism theorem, proves $\R[x]/(x^2 + 1) \cong \C$?

How many residues $x \in \Z/35\Z$ satisfy $x^2 = x$? (Use the Chinese remainder theorem.)

Which Lean statement formalises “for a commutative ring $R$ and an ideal $I$, the quotient $R/I$ is a field if and only if $I$ is maximal”?

A student says: “$\Z/2\Z$ is a subgroup of $\Z$, namely $\{0, 1\}$.” What is wrong?

Write cosets(n, k) returning the cosets of the subgroup $H = \langle k \rangle$ of $\Z/n\Z$, as a list of sorted lists, ordered by their smallest element. Then write coset_sum(n, k, a, b) returning the coset (as a sorted list) of $a + b$. Check that the coset of $a + b$ does not change when $a$ or $b$ is replaced by another element of its coset.