8 A group \(D\) of order 10 is generated by the elements \(a\) and \(r\), with the properties \(a ^ { 2 } = e , r ^ { 5 } = e\) and \(r ^ { 4 } a = a r\), where \(e\) is the identity. Part of the operation table is shown below.
| \(e\) | \(а\) | \(r\) | \(r ^ { 2 }\) | \(r ^ { 3 }\) | \(r ^ { 4 }\) | ar | \(a r ^ { 2 }\) | \(a r ^ { 3 }\) | \(a r ^ { 4 }\) |
| \(e\) | \(e\) | \(а\) | \(r\) | \(r ^ { 2 }\) | \(r ^ { 3 }\) | \(r ^ { 4 }\) | ar | \(a r ^ { 2 }\) | \(a r ^ { 3 }\) | \(a r ^ { 4 }\) |
| \(а\) | \(а\) | \(e\) | ar | \(a r ^ { 2 }\) | \(a r ^ { 3 }\) | \(a r ^ { 4 }\) | | | | |
| \(r\) | r | | \(r ^ { 2 }\) | \(r ^ { 3 }\) | \(r ^ { 4 }\) | \(e\) | | | | |
| \(r ^ { 2 }\) | \(r ^ { 2 }\) | | \(r ^ { 3 }\) | \(r ^ { 4 }\) | \(e\) | \(r\) | | | | |
| \(r ^ { 3 }\) | \(r ^ { 3 }\) | | \(r ^ { 4 }\) | \(e\) | \(r\) | \(r ^ { 2 }\) | | | | |
| \(r ^ { 4 }\) | \(r ^ { 4 }\) | ar | \(e\) | \(r\) | \(r ^ { 2 }\) | \(r ^ { 3 }\) | | | | |
| ar | ar | | \(a r ^ { 2 }\) | \(a r ^ { 3 }\) | \(a r ^ { 4 }\) | \(а\) | | | | |
| \(a r ^ { 2 }\) | \(a r ^ { 2 }\) | | \(a r ^ { 3 }\) | \(a r ^ { 4 }\) | \(a\) | ar | | | T | |
| \(a r ^ { 3 }\) | \(a r ^ { 3 }\) | | \(a r ^ { 4 }\) | \(а\) | ar | \(a r ^ { 2 }\) | | | | |
| \(a r ^ { 4 }\) | \(a r ^ { 4 }\) | | \(а\) | ar | \(a r ^ { 2 }\) | \(a r ^ { 3 }\) | | | | |
- Give a reason why \(D\) is not commutative.
- Write down the orders of any possible proper subgroups of \(D\).
- List the elements of a proper subgroup which contains
(a) the element \(a\),
(b) the element \(r\). - Determine the order of each of the elements \(r ^ { 3 }\), \(a r\) and \(a r ^ { 2 }\).
- Copy and complete the section of the table marked \(\mathbf { E }\), showing the products of the elements \(a r , a r ^ { 2 } , a r ^ { 3 }\) and \(a r ^ { 4 }\).