OCR MEI FP3 2013 June — Question 4

Exam BoardOCR MEI
ModuleFP3 (Further Pure Mathematics 3)
Year2013
SessionJune
TopicGroups
4
  1. The composition table for a group \(G\) of order 8 is given below.
    \(a\)\(b\)\(c\)\(d\)\(e\)\(f\)\(g\)\(h\)
    \(a\)\(c\)\(e\)\(b\)\(f\)\(a\)\(h\)\(d\)\(g\)
    \(b\)\(e\)\(c\)\(a\)\(g\)\(b\)\(d\)h\(f\)
    \(c\)\(b\)\(a\)\(e\)\(h\)\(c\)\(g\)\(f\)\(d\)
    \(d\)\(f\)\(g\)\(h\)\(a\)\(d\)\(c\)\(e\)\(b\)
    \(e\)\(a\)\(b\)\(c\)\(d\)\(e\)\(f\)\(g\)\(h\)
    \(f\)\(h\)\(d\)\(g\)\(c\)\(f\)\(b\)\(a\)\(e\)
    \(g\)\(d\)\(h\)\(f\)\(e\)\(g\)\(a\)\(b\)\(c\)
    \(h\)\(g\)\(f\)\(d\)\(b\)\(h\)\(e\)\(c\)\(a\)
    1. State which is the identity element, and give the inverse of each element of \(G\).
    2. Show that \(G\) is cyclic.
    3. Specify an isomorphism between \(G\) and the group \(H\) consisting of \(\{ 0,2,4,6,8,10,12,14 \}\) under addition modulo 16 .
    4. Show that \(G\) is not isomorphic to the group of symmetries of a square.
  2. The set \(S\) consists of the functions \(\mathrm { f } _ { n } ( x ) = \frac { x } { 1 + n x }\), for all integers \(n\), and the binary operation is composition of functions.
    1. Show that \(\mathrm { f } _ { m } \mathrm { f } _ { n } = \mathrm { f } _ { m + n }\).
    2. Hence show that the binary operation is associative.
    3. Prove that \(S\) is a group.
    4. Describe one subgroup of \(S\) which contains more than one element, but which is not the whole of \(S\).
This paper (5 questions)
View full paper
Q1 Q2 Q3 Q4 Q5