OCR Further Additional Pure 2022 June — Question 8

Exam BoardOCR
ModuleFurther Additional Pure (Further Additional Pure)
Year2022
SessionJune
TopicGroups
8
  1. Explain why all groups of even order must contain at least one self-inverse element (that is, an element of order 2).
  2. Prove that any group, in which every (non-identity) element is self-inverse, is abelian.
  3. A student believes that, if \(x\) and \(y\) are two distinct, non-identity, self-inverse elements of a group, then the element \(x y\) is also self-inverse. The table shown here is the Cayley table for the non-cyclic group of order 6, having elements \(i , a , b , c , d\) and \(e\), where \(i\) is the identity.
    \(i\)\(a\)\(b\)\(c\)\(d\)\(e\)
    \(i\)\(i\)\(a\)\(b\)\(c\)\(d\)\(e\)
    \(a\)\(a\)\(i\)\(d\)\(e\)\(b\)\(c\)
    \(b\)\(b\)\(e\)\(i\)\(d\)\(c\)\(a\)
    \(c\)\(c\)\(d\)\(e\)\(i\)\(a\)\(b\)
    \(d\)\(d\)\(c\)\(a\)\(b\)\(e\)\(i\)
    \(e\)\(e\)\(b\)\(c\)\(a\)\(i\)\(d\)
    By considering the elements of this group, produce a counter-example which proves that this student is wrong.
  4. A group \(G\) has order \(4 n + 2\), for some positive integer \(n\), and \(i\) is the identity element of \(G\). Let \(x\) and \(y\) be two distinct, non-identity, self-inverse elements of \(G\). By considering the set \(\mathrm { H } = \{ \mathrm { i } , \mathrm { x } , \mathrm { y } , \mathrm { xy } \}\), prove by contradiction that not all elements of \(G\) are self-inverse.
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