OCR Further Additional Pure 2023 June — Question 5

Exam BoardOCR
ModuleFurther Additional Pure (Further Additional Pure)
Year2023
SessionJune
TopicGroups
5
  1. The group \(G\) consists of the set \(S = \{ 1,9,17,25 \}\) under \(\times _ { 32 }\), the operation of multiplication modulo 32.
    1. Complete the Cayley table for \(G\) given in the Printed Answer Booklet.
    2. Up to isomorphisms, there are only two groups of order 4.
      • \(C _ { 4 }\), the cyclic group of order 4
  2. \(K _ { 4 }\), the non-cyclic (Klein) group of order 4
    3. State, with justification, to which of these two groups \(G\) is isomorphic.
    1. List the odd quadratic residues modulo 32.
    2. Given that \(n\) is an odd integer, prove that \(n ^ { 6 } + 3 n ^ { 4 } + 7 n ^ { 2 } \equiv 11 ( \bmod 32 )\).
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