4 A group \(G\), of order 8, is generated by the elements \(a , b , c . G\) has the properties
$$a ^ { 2 } = b ^ { 2 } = c ^ { 2 } = e , \quad a b = b a , \quad b c = c b , \quad c a = a c ,$$
where \(e\) is the identity.
- Using these properties and basic group properties as necessary, prove that \(a b c = c b a\).
The operation table for \(G\) is shown below.
| \(e\) | \(a\) | \(b\) | \(c\) | \(b c\) | ca | \(a b\) | \(a b c\) |
| \(e\) | \(e\) | \(a\) | \(b\) | \(c\) | \(b c\) | ca | \(a b\) | \(a b c\) |
| \(a\) | \(a\) | \(e\) | \(a b\) | ca | \(a b c\) | \(c\) | \(b\) | \(b c\) |
| \(b\) | \(b\) | \(a b\) | \(e\) | \(b c\) | \(c\) | \(a b c\) | \(a\) | ca |
| c | \(c\) | ca | \(b c\) | \(e\) | \(b\) | \(a\) | \(a b c\) | \(a b\) |
| \(b c\) | \(b c\) | \(a b c\) | \(c\) | \(b\) | \(e\) | \(a b\) | ca | \(a\) |
| ca | ca | c | \(a b c\) | \(a\) | \(a b\) | \(e\) | \(b c\) | \(b\) |
| \(a b\) | \(a b\) | \(b\) | \(a\) | \(a b c\) | ca | bc | \(e\) | \(c\) |
| \(a b c\) | \(a b c\) | \(b c\) | ca | \(a b\) | \(a\) | \(b\) | \(c\) | \(e\) |
- List all the subgroups of order 2 .
- List five subgroups of order 4.
- Determine whether all the subgroups of \(G\) which are of order 4 are isomorphic.