Let $G, H$ be groups and let $\varphi: G \rightarrow H$ be a function. What does it mean to say that $\varphi$ is a homomorphism with kernel $K$ ? Show that if $K=\{e, \xi\}$ has order 2 then $x^{-1} \xi x=\xi$ for each $x \in G$. [If you use any general results about kernels of homomorphisms, then you should prove them.]

Which of the following four statements are true, and which are false? Justify your answers.

(a) There is a homomorphism from the orthogonal group $\mathrm{O}(3)$ to a group of order 2 with kernel the special orthogonal group $\mathrm{SO}(3)$.

(b) There is a homomorphism from the symmetry group $S_{3}$ of an equilateral triangle to a group of order 2 with kernel of order 3 .

(c) There is a homomorphism from $\mathrm{O}(3)$ to $\mathrm{SO}(3)$ with kernel of order 2 .

(d) There is a homomorphism from $S_{3}$ to a group of order 3 with kernel of order 2 .