Elements of semigroup commuting with subgroup

[mnb96]

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 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?

 

[fresh_42]

You can say that s is in the centralizer of G in S.

 

[WWGD]

And the term extends to subsets in general, not just to subgroups.

 

[mnb96]

Thanks a lot for your answers, especially for pointing out the centralizer.

 

[WWGD]

Thanks a lot for your answers, especially for pointing out the centralizer.

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