Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 2 / 9

The kernel of a group homomorphism isThe homomorphism sends identity to identity and inverses to inverses. Thus $e_G\in K$, and if $k_1,k_2\in K$, then $\Phi (k_1{k}_2^{-1})=e_{G^{\prime }}$, so $K$ is a subgroup by the subgroup test.

For any $g\in G$ and $k\in K$,Hence $gK{g}^{-1}\subseteq K$. Applying the same inclusion with $g^{-1}$ gives equality, provingThis normality argument does not require either group to be finite.

The conjugacy class of $g$ isIf $N$ is a normal subgroup and $g\in N$, then normality gives $hg{h}^{-1}\in N$ for every $h$. Thus the whole conjugacy class of each member of $N$ is contained in $N$, andConjugacy classes partition the group, so this is a union of complete disjoint classes, never selected fractions of classes.

The cycle type records the lengths of the disjoint cycles in a permutation, including fixed points as length-one cycles. The possibilities for three objects areThese exhaust the six elements of the symmetric group $S_3$, denoted ${\Sigma }_3$ in the source. They are also the conjugacy classes of the symmetric group on three letters.

The kernel is a normal subgroup, so it contains the identity class and is a union of the classes listed in part (c). Its candidate orders are therefore $1$, $1+3=4$, $1+2=3$ and $1+3+2=6$. Lagrange's theorem excludes four, since it does not divide six. The remaining possibilities areAll three are normal: the first and last trivially, while $A_3$ is the kernel of the map giving the sign of a permutation, or the index-two subgroup generated by a 3-cycle.

Because $\Phi$ is onto, the first isomorphism theorem gives $G^{\prime }\cong S_3/K$. Explicitly, the map $gK\mapsto \Phi (g)$ is well defined, multiplicative and surjective; equality of its values means $h^{-1}g\in K$, so it is also injective. ThusEach is realized by the identity map, the sign map and the trivial map, respectively. This completes the classification of normal subgroups and quotients of the symmetric group on three letters.