8 A non-commutative multiplicative group \(G\) of order eight has the elements
$$\left\{ e , a , a ^ { 2 } , a ^ { 3 } , b , a b , a ^ { 2 } b , a ^ { 3 } b \right\}$$
where \(e\) is the identity and \(a ^ { 4 } = b ^ { 2 } = e\).
- Show that \(b a \neq a ^ { n }\) for any integer \(n\).
- Prove, by contradiction, that \(b a \neq a ^ { 2 } b\) and also that \(b a \neq a b\). Deduce that \(b a = a ^ { 3 } b\).
- Prove that \(b a ^ { 2 } = a ^ { 2 } b\).
- Construct group tables for the three subgroups of \(G\) of order four.
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