Order of elements

2 The elements of a group \(G\) are the complex numbers \(a + b \mathrm { i }\) where \(a , b \in \{ 0,1,2,3,4 \}\). These elements are combined under the operation of addition modulo 5 .

1. State the identity element and the order of \(G\).
2. Write down the inverse of \(2 + 4 \mathrm { i }\).
3. Show that every non-zero element of \(G\) has order 5 .

\(Q\) is a multiplicative group of order 12.

1. Two elements of \(Q\) are \(a\) and \(r\). It is given that \(r\) has order 6 and that \(a^2 = r^3\). Find the orders of the elements \(a\), \(a^2\), \(a^3\) and \(r^2\). [4]

The table below shows the number of elements of \(Q\) with each possible order. \(G\) and \(H\) are the non-cyclic groups of order 4 and 6 respectively.

1. Construct two tables, similar to the one above, to show the number of elements with each possible order for the groups \(G\) and \(H\). Hence explain why there are no non-cyclic proper subgroups of \(Q\). [5]