OCR FP3 2008 January — Question 1

Exam BoardOCR
ModuleFP3 (Further Pure Mathematics 3)
Year2008
SessionJanuary
TopicGroups
1
  1. A group \(G\) of order 6 has the combination table shown below.
    \(e\)\(a\)\(b\)\(p\)\(q\)\(r\)
    \(e\)\(e\)\(a\)\(b\)\(p\)\(q\)\(r\)
    \(a\)\(a\)\(b\)\(e\)\(r\)\(p\)\(q\)
    \(b\)\(b\)\(e\)\(a\)\(q\)\(r\)\(p\)
    \(p\)\(p\)\(q\)\(r\)\(e\)\(a\)\(b\)
    \(q\)\(q\)\(r\)\(p\)\(b\)\(e\)\(a\)
    \(r\)\(r\)\(p\)\(q\)\(a\)\(b\)\(e\)
    1. State, with a reason, whether or not \(G\) is commutative.
    2. State the number of subgroups of \(G\) which are of order 2 .
    3. List the elements of the subgroup of \(G\) which is of order 3 .
  2. A multiplicative group \(H\) of order 6 has elements \(e , c , c ^ { 2 } , c ^ { 3 } , c ^ { 4 } , c ^ { 5 }\), where \(e\) is the identity. Write down the order of each of the elements \(c ^ { 3 } , c ^ { 4 }\) and \(c ^ { 5 }\).
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