6 The group \(G\) consists of the set \(\{ 3,6,9,12,15,18,21,24,27,30,33,36 \}\) under \(\times _ { 39 }\), the operation of multiplication modulo 39.
- List the possible orders of proper subgroups of \(G\), justifying your answer.
- List the elements of the subset of \(G\) generated by the element 3 .
- State the identity element of \(G\).
- Determine the order of the element 18 .
- Find the two elements \(g _ { 1 }\) and \(g _ { 2 }\) in \(G\) which satisfy \(g \times { } _ { 39 } g = 3\).
The group \(H\) consists of the set \(\{ 1,2,3,4,5,6,7,8,9,10,11,12 \}\) under \(\times _ { 13 }\), the operation of multiplication modulo 13. You are given that \(G\) is isomorphic to \(H\).
A student states that \(G\) is isomorphic to \(H\) because each element \(3 x\) in \(G\) maps directly to the element \(x\) in \(H\) (for \(x = 1,2,3,4,5,6,7,8,9,10,11,12\) ).
- Explain why this student is incorrect.