1.13: Exercises
- Page ID
- 180042
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[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 \(G\).
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Show that every group of exponent \(2\) is commutative.
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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 \(\mathbb{Z}{}\).