Groups with generators and relations

8 A multiplicative group \(Q\) of order 8 has elements \(\left\{ e , p , p ^ { 2 } , p ^ { 3 } , a , a p , a p ^ { 2 } , a p ^ { 3 } \right\}\), where \(e\) is the identity. The elements have the properties \(p ^ { 4 } = e\) and \(a ^ { 2 } = p ^ { 2 } = ( a p ) ^ { 2 }\).

1. Prove that \(a = p a p\) and that \(p = a p a\).
2. Find the order of each of the elements \(p ^ { 2 } , a , a p , a p ^ { 2 }\).
3. Prove that \(\left\{ e , a , p ^ { 2 } , a p ^ { 2 } \right\}\) is a subgroup of \(Q\).
4. Determine whether \(Q\) is a commutative group.

A group \(D\) of order 10 is generated by the elements \(a\) and \(r\), with the properties \(a^2 = e\), \(r^5 = e\) and \(r^4a = ar\), where \(e\) is the identity. Part of the operation table is shown below. \includegraphics{figure_1}

1. Give a reason why \(D\) is not commutative. [1]
2. Write down the orders of any possible proper subgroups of \(D\). [2]
3. List the elements of a proper subgroup which contains
   1. the element \(a\), [1]
   2. the element \(r\). [1]
4. Determine the order of each of the elements \(r^3\), \(ar\) and \(ar^2\). [4]
5. Copy and complete the section of the table marked E, showing the products of the elements \(ar\), \(ar^2\), \(ar^3\) and \(ar^4\). [5]

A group \(G\), of order 8, is generated by the elements \(a\), \(b\), \(c\). \(G\) has the properties $$a^2 = b^2 = c^2 = e, \quad ab = ba, \quad bc = cb, \quad ca = ac,$$ where \(e\) is the identity.

1. Using these properties and basic group properties as necessary, prove that \(abc = cba\). [2]

The operation table for \(G\) is shown below.

1. List all the subgroups of order 2. [2]
2. List five subgroups of order 4. [3]
3. Determine whether all the subgroups of \(G\) which are of order 4 are isomorphic. [2]