9 The group \(\left( C , + _ { 4 } \right)\) contains the elements \(0,1,2\) and 3
9
- Show that \(C\) is a cyclic group.
9
- (ii) State the group of symmetries of a regular polygon that is isomorphic to \(C\)
9 - The group ( \(V , \otimes\) ) contains the elements (1, 1), (1, -1), (-1, 1) and (-1, -1) The binary operation \(\otimes\) between elements of \(V\) is defined by
$$( a , b ) \otimes ( c , d ) = ( a \times c , b \times d )$$
9
- Find the element in \(V\) that is the inverse of \(( - 1,1 )\)
Fully justify your answer.
[0pt]
[2 marks]
9
- (ii) Determine, with a reason, whether or not \(C \cong V\)
\(\mathbf { 9 }\) (c) The group \(G\) has order 16
Rachel claims that as \(1,2,4,8\) and 16 are the only factors of 16 then, by Lagrange's theorem, the group \(G\) will have exactly 5 distinct subgroups, including the trivial subgroup and \(G\) itself.
Comment on the validity of Rachel's claim.
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