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Let $G$ be a group. Let $S \subset G$. Consider the set of all $x \in G$ such that $xS = S$. What is this unique largest subgroup of $G$ preserving $S$ under left-multiplication called?
(As for the plural used in the title, there is an analogous subgroup for right-multiplication.)

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    $\begingroup$ Why isn't this just the subgroup of all $k$ such that $kS = S$ (so the stabilizer of $S$ in $2^G$)? $\endgroup$
    – Qiaochu Yuan
    Commented Nov 23, 2013 at 1:42
  • $\begingroup$ Yes. Yes it is. I logged in because I just realized this. $\endgroup$
    – DavidLHarden
    Commented Nov 23, 2013 at 8:37
  • $\begingroup$ (And I removed superfluous elements of my reasoning.) $\endgroup$
    – DavidLHarden
    Commented Nov 23, 2013 at 8:45

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It is the set-wise stabilizer of $S$ under the left regular action.

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