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## Main Question or Discussion Point

Hello,

Suppose we have a semigroup S with a subgroup G≤S.

Assume there is an element s∈S that commutes with all the elements in G. Does this statement implies (or is equivalent to) another statement?

If hypothetically the element

Suppose we have a semigroup S with a subgroup G≤S.

Assume there is an element s∈S that commutes with all the elements in G. Does this statement implies (or is equivalent to) another statement?

If hypothetically the element

*s*would have been in G, then we could have said that*s*was an element of the*center*of G, but since*s*is not contained in G then I suspect we can't say much more than just saying that it commutes with G, or can we?