OCR MEI FP3 2012 June — Question 4

Exam BoardOCR MEI
ModuleFP3 (Further Pure Mathematics 3)
Year2012
SessionJune
TopicGroups
4
  1. Show that the set \(P = \{ 1,5,7,11 \}\), under the binary operation of multiplication modulo 12, is a group. You may assume associativity. A group \(Q\) has identity element \(e\). The result of applying the binary operation of \(Q\) to elements \(x\) and \(y\) is written \(x y\), and the inverse of \(x\) is written \(x ^ { - 1 }\).
  2. Verify that the inverse of \(x y\) is \(y ^ { - 1 } x ^ { - 1 }\). Three elements \(a , b\) and \(c\) of \(Q\) all have order 2, and \(a b = c\).
  3. By considering the inverse of \(c\), or otherwise, show that \(b a = c\).
  4. Show that \(b c = a\) and \(a c = b\). Find \(c b\) and \(c a\).
  5. Complete the composition table for \(R = \{ e , a , b , c \}\). Hence show that \(R\) is a subgroup of \(Q\) and that \(R\) is isomorphic to \(P\). The group \(T\) of symmetries of a square contains four reflections \(A , B , C , D\), the identity transformation \(E\) and three rotations \(F , G , H\). The binary operation is composition of transformations. The composition table for \(T\) is given below.
    A\(B\)\(C\)D\(E\)\(F\)\(G\)\(H\)
    AE\(G\)\(H\)\(F\)\(A\)D\(B\)\(C\)
    BGE\(F\)\(H\)\(B\)CAD
    C\(F\)HEGCAD\(B\)
    D\(H\)\(F\)\(G\)E\(D\)\(B\)C\(A\)
    EA\(B\)CD\(E\)\(F\)\(G\)\(H\)
    FCD\(B\)A\(F\)G\(H\)\(E\)
    \(G\)B\(A\)\(D\)C\(G\)HE\(F\)
    \(H\)DCAB\(H\)E\(F\)G
  6. Find the order of each element of \(T\).
  7. List all the proper subgroups of \(T\).
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