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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".2012-06-27
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    $S$ 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.2012-06-27
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    Is it clear that $S$ is well defined? Let $G=(a,b)$ with $a,b \in S_3$. Then if $e$ is the identity of $S_3$, $(a,e)$ commutes with $(e,b)$, but there are other elements it doesn't commute with. So $S=\{(e,e),(a,e),(e,b)\}$ is maximal, but $a,b$ can be any elements of $S_3$. You could also have $S=\{(e,e),(a,b)\}$ with $a,b$ non-identity, which I believe is also maximal.2012-06-27
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    @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.2012-06-27
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    @ArturoMagidin: OK. There could be a number of options for $S$ then.2012-06-27
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    @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.2012-06-27
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    @ArturoMagidin Yes, that's exactly it.2012-06-27
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    @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".2012-06-27
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    Related (but not exactly the same that you asked): http://en.wikipedia.org/wiki/Center_(group_theory)2015-04-06

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