1.13: Exercises

[x1] Show that the quaternion group has only one element of order $2$, and that it commutes with all elements of $Q$. Deduce that $Q$ is not isomorphic to $D_{4}$, and that every subgroup of $Q$ is normal.¹²

[x2] Consider the elements

$$a=% \begin{pmatrix} 0 & -1\\ 1 & \hfill\hfill0 \end{pmatrix} \quad b=% \begin{pmatrix} \hfill0 & \hfill1\\ -1 & -1 \end{pmatrix} \nonumber$$

in $\GL_{2}(\mathbb{Z})$. Show that $a^{4}=1$ and $b^{3}=1$, but that $ab$ has infinite order, and hence that the group $\langle a,b\rangle$ is infinite.

[x3] Show that every finite group of even order contains an element of order $2$.

[x4d]Let $n=n_{1}+\cdots+n_{r}$ be a partition of the positive integer $n$. Use Lagrange’s theorem to show that $n!$ is divisible by $\prod \nolimits_{i=1}^{r}n_{i}!$.

[x4] Let $N$ be a normal subgroup of $G$ of index $n$. Show that if $g\in G$, then $g^{n}\in N$. Give an example to show that this may be false when the subgroup is not normal.

[x4a] A group $G$ is said to have finite exponent if there exists an $m>0$ such that $a^{m}=e$ for every $a$ in $G$; the smallest such $m$ is then called the exponent of $G$.

1. Show that every group of exponent $2$ is commutative.

2. Show that, for an odd prime $p$, the group of matrices

   $$\left\{ \begin{pmatrix} 1 & a & b\\ 0 & 1 & c\\ 0 & 0 & 1 \end{pmatrix} \,\, \middle|\,\, a,b,c\in\mathbb{F}{}_{p}\right\} \nonumber$$

   has exponent $p$, but is not commutative.

[x4b]Two subgroups $H$ and $H^{\prime}$ of a group $G$ are said to be commensurable if $H\cap H^{\prime}$ is of finite index in both $H$ and $H^{\prime}$. Show that commensurability is an equivalence relation on the subgroups of $G$.

[x4c]Show that a nonempty finite set with an associative binary operation satisfying the cancellation laws is a group.

[x4e]Let $G$ be a set with an associative binary operation. Show that if left multiplication $x\mapsto ax$ by every element $a$ is bijective and right multiplication by some element is injective, then $G$ is a group. Give an example to show that the second condition is needed.

[x0]Show that a commutative monoid $M$ is a submonoid of a commutative group if and only if cancellation holds in $M$:

$$mn=m^{\prime}n\implies m=m^{\prime}. \nonumber$$

Hint: The group is constructed from $M$ as $\mathbb{Q}{}$ is constructed from $\mathbb{Z}{}$.