Let G be a group and g and h be elements in G. If g commutes with all conjugates of g and h commutes with all conjugates of h, can one conclude that gh commutes with all conjugates of gh?


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Question about some element in a group commutes with its all conjuagates.
No; the smallest counterexamples are given by the groups SmallGroup(54,5) and SmallGroup(54,6) (in GAP's SmallGroups library); these are groups of the form
$$G\_1 = ((3 \times 3) : 3) : 2 \text{ and } G\_2 = (9 : 3) : 2.$$
In both cases, there are 15 elements with the given property (i.e. that they commute with all their conjugates), so they can't possibly form a subgroup.


