Question 6 (Unit B4) — 12 marks
Let G be a group, and let $a \in G$. Let H be the subset of G given by
$H = \{x \in G : ax^2 = x^2a\}$.
(a) Show that properties SG2 and SG3 in the subgroup test hold for H as a
subset of G. Carefully justify the steps of your arguments.
(b) Show that property SG1 in the subgroup test does not necessarily hold
for H as a subset of G, by considering the case
$G = S_4$, $a = (1\ 2)$,
and using the permutations
$x = (1\ 3)$, $y = (1\ 4)$.
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