Conjugacy Classes and Centralisers
By definition, the centre is the set of elements for which the conjugacy class of each element is the element itself, i.e. ccl(g) = {g}.
The centre is also the intersection of all the centralisers of each element of G. As centralisers are subgroups, this again shows that the centre is a subgroup.
Read more about this topic: Center (group Theory)
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