Algebra Solutions

Find a presentation for ZZ/(n)

Find a set of generators and relations for $\mathbb{Z}/(n)$ .

We saw in a previous exercise that the distinct elements of $\mathbb{Z}/(n)$ are precisely $\overline{0}, \overline{1}, \ldots \overline{n-1}$ . Note that these are precisely the multiples of $\overline{1}$ ; hence $\mathbb{Z}/(n)$ is generated by $\overline{1}$ . Since $|\overline{1}| = n$ , we have $\mathbb{Z}/(n) = \langle x \ |\ x^n = 1 \rangle$ .

Generating sets of the integers as an additive group

Find a set of generators for $\mathbb{Z}$ .

Every integer can be written as a finite sum of 1s, so we have $\mathbb{Z} = \langle 1 \rangle$ . In particular, $\mathbb{Z}$ is a cyclic group.
Also, suppose $a$ and $b$ are integers with $\mathsf{gcd}(a,b) = 1$ . Then there exist integers $x,y$ with $ax + by = 1$ , so we also have $\mathbb{Z} = \langle a, b \rangle$ .

Compute the number of orientation-preserving isometries of an icosahedron

Let $G$ be the group of rigid motions (i.e., orientation preserving isometries) of an icosahedron in $\mathbb{R}^3$ . Show that $|G| = 60$ . (Find the number of positions to which an adjacent pair of vertices can be sent and argue that fixing one edge uniquely determines the isometry.)

For reference, consider the following diagram of an icosahedron.

Let $\theta$ be an orientation-preserving isometry of the icosahedron; that is, if the vertices of a face, read clockwise from outside the figure, are $XYZ$ , then $\theta(X) \theta(Y) \theta(Z)$ are the vertices of the corresponding face, read clockwise from outside the figure, of the isometric copy.
There are 12 possibilities for $\theta(1)$ . Once $\theta(1)$ is chosen, there are 5 possibilities for $\theta(2)$ . Once these are chosen, the rest of $\theta$ is determined uniquely by orientation. Thus there are $12 \cdot 5 = 60$ possibilities for $\theta$ , all distinct. $\blacksquare$

Find the number of orientation-preserving isometries of a dodecahedron

Let $G$ be the group of rigid motions (i.e., orientation preserving isometries) of a dodecahedron in $\mathbb{R}^3$ . Show that $|G| = 60$ .

For reference, consider the following diagram of a dodecahedron.

Let $\theta$ be an orientation-preserving isometry of the dodecahedron; that is, if the vertices of a face, read clockwise from outside the figure, are $VWXYZ$ , then $\theta(V) \theta(W) \theta(X) \theta(Y) \theta(Z)$ are the vertices of the corresponding face, read clockwise from outside the figure, of the isometric copy.
There are 20 possibilities for $\theta(1)$ . Once $\theta(1)$ is chosen, there are 3 possibilities for $\theta(2)$ . Once these are chosen, the rest of $\theta$ is determined uniquely by orientation. Thus there are $20 \cdot 3 = 60$ possibilities for $\theta$ , all distinct.

Find the number of orientation-preserving isometries of an octahedron

Let $G$ be the group of rigid motions (i.e., orientation preserving isometries) of an octahedron in $\mathbb{R}^3$ . Show that $|G| = 24$ . (Find the number of positions to which an adjacent pair of vertices can be sent and argue that fixing one edge uniquely determines the isometry.)

For reference, consider the following diagram of an octahedron.

Let $\theta$ be an orientation-preserving isometry of the octahedron; that is, if the vertices of a face, read clockwise from outside the figure, are $XYZ$ , then $\theta(X) \theta(Y) \theta(Z)$ are the vertices of the corresponding face, read clockwise from outside the figure, of the isometric copy.
There are 6 possibilities for $\theta{1}$ . Once $\theta(1)$ is chosen, there are 4 possibilities for $\theta(2)$ . Then the rest of $\theta$ is determined uniquely by orientation. Thus there are $6 \cdot 4 = 24$ possibilities for $\theta$ , add distinct.