Supplementary Problems

Let $A, B$ be $3 \times 3$ matrices over $\mathbb {R}$. We define the composition

Does this composition define a group? The neutral element is $0_{3}$.

Consider the set

We define the composition $(a,b) \bullet (c,d) := (ac, (b+d) \bmod 2\pi )$. Show that this composition defines a group.

Let $m \in \mathbb {Z}$, $m \geq 2$, and let $G(m)$ be the group of $2 \times 2$ matrices of determinant $\pm 1$ and with entries in the ring $\mathbb {Z}/m\mathbb {Z}$, i.e.

Show that the set $G(m)$ forms a finite group under matrix multiplication.

Show that there is only one group of order three. Is the group commutative? Is the group isomorphic to a subgroup of $S_{3}$?

Let $\phi : G \rightarrow H$ be a homomorphism between the groups $G$ and $H$. Show that $\ker (\phi )$ is a normal subgroup of $G$.

Let $\alpha , \beta , \phi \in \mathbb {R}$ and $\alpha , \beta \neq 0$. Consider the matrices

Do the matrices form a group under matrix multiplication?

Let $S_{n}$ be the group of all permutations of $n$ objects. Show that every finite group is isomorphic to a group of permutations.

Let $G$ be a group. An automorphism of $G$ is an isomorphism sending $G$ onto itself. Show that the set $\operatorname {Aut}(G)$ of automorphisms of $G$ is a group with respect to the operation of composition of automorphisms.