Given that \(\sqrt { 7 }\) is an irrational number, prove that \(a ^ { 2 } - 7 b ^ { 2 } \neq 0\) for all \(a , b \in \mathbb { Q }\) where \(a\) and \(b\) are not both 0 .
A set \(G\) is defined by \(G = \{ a + b \sqrt { 7 } : a , b \in \mathbb { Q } , a\) and \(b\) not both \(0 \}\).
Prove that \(G\) is a group under multiplication. (You may assume that multiplication is associative.)
A subset \(H\) of \(G\) is defined by \(H = \{ 1 + c \sqrt { 7 } : c \in \mathbb { Q } \}\).
Determine whether or not \(H\) is a subgroup of ( \(G , \times\) ).
Using \(( G , \times )\), prove by counter-example that the statement 'An infinite group cannot have a non-trivial subgroup of finite order' is false.