Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 3 / 6D (Groups)

With $D_8=⟨r,s:r^4=s^2=1,srs=r^{-1}⟩$, its normal subgroups areThe last three proper nontrivial examples have index two; the four individual reflection subgroups are not normal.

If $K◃G$, the quotient map $G\to G/K$ is surjective with kernel $K$. Conversely, every kernel is normal because $\varphi (gk{g}^{-1})=\varphi (g)1\varphi (g{)}^{-1}=1$.

False. The quaternion group $Q_8$ is nonabelian, but each of its subgroups is normal: its nontrivial proper subgroups are its centre $\{\pm 1\}$ and the three cyclic subgroups of order four, all of index two.

For $g\in G$ and $x\in {\varphi }^{-1}(N)$,so the preimage is normal.

Given $h\in H$, choose $g\in G$ with $\varphi (g)=h$. For $k\in K$,because $K$ is normal. Thus $\varphi (K)◃H$.