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Let $G$ be a group and $S$ its subset. I would like to consider the following condition on $S$.

For every $x,y\in S,$ we have $xy=yx.$

This is trivially equivalent to $S\subseteq C(S).$

The same condition can be formulated for a semigroup, and if we define the centralizer of a subset of a semigroup in the same way as for a group, then the equivalence still obviously holds.

I would like to know if there is a name for this condition.

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I would call it "commutative subset", or a set of "pairwise commuting elements". – Arturo Magidin Jun 27 '12 at 0:35
$S$ is a subset of an abelian subgroup, namely $\langle S \rangle$. You could even call it a generating set of an abelian subgroup. These are both defining conditions. – Jack Schmidt Jun 27 '12 at 0:56
@Ross: I didn't understand the question as asking for a maximal $S$. Rather, you have a set $S$, which just happens to satisfy the condition that any two elements of $S$ commute with each other. – Arturo Magidin Jun 27 '12 at 2:53
@Ross: Yes; if $S$ were a subsemigroup/subgroup, you would call it an abelian/commutative subsemigroup/subgroup. But I'm guessing that ymar is wondering what to call it if it is just a set, given that "abelian subset" is not a term one hears. As Jack says, one could say it is a subset of an abelian subgroup. – Arturo Magidin Jun 27 '12 at 2:57
@ymar: I don't think there is any special name. You could call it a "set of commuting elements", or as Jack notes, "contained in a commutative subgroup/subsemigroup". – Arturo Magidin Jun 27 '12 at 3:45

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