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Post Closed as "Not suitable for this site" by Benoît Kloeckner, Daniel Moskovich, Chris Godsil, Carlo Beenakker, David White
re-inserted line break before parenthetical remark
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DavidLHarden
DavidLHarden
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Let $G$ be a group. Let $S \subset G$. Consider the partially ordered set of subgroupsall $K < G$$x \in G$ such that $k \cdot S = S$ for all $k \in K$.
If $K_{i}$ is a totally ordered set of subgroups of $G$ with this property, then $\bigcup_{i} K_{i}$ is also a subgroup with this property. Therefore, by Zorn's Lemma, the partially ordered set of subgroups of $G$ preserving $S$ by left-multiplication has a maximal element.
Furthermore, this maximal element is unique: if $M$ and $K$ are subgroups preserving $S$ by left-multiplication, then so is their join $<M, K>$$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.)

Let $G$ be a group. Let $S \subset G$. Consider the partially ordered set of subgroups $K < G$ such that $k \cdot S = S$ for all $k \in K$.
If $K_{i}$ is a totally ordered set of subgroups of $G$ with this property, then $\bigcup_{i} K_{i}$ is also a subgroup with this property. Therefore, by Zorn's Lemma, the partially ordered set of subgroups of $G$ preserving $S$ by left-multiplication has a maximal element.
Furthermore, this maximal element is unique: if $M$ and $K$ are subgroups preserving $S$ by left-multiplication, then so is their join $<M, K>$.
  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.)

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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DavidLHarden
DavidLHarden
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What are these subgroups called?

Let $G$ be a group. Let $S \subset G$. Consider the partially ordered set of subgroups $K < G$ such that $k \cdot S = S$ for all $k \in K$.
If $K_{i}$ is a totally ordered set of subgroups of $G$ with this property, then $\bigcup_{i} K_{i}$ is also a subgroup with this property. Therefore, by Zorn's Lemma, the partially ordered set of subgroups of $G$ preserving $S$ by left-multiplication has a maximal element.
Furthermore, this maximal element is unique: if $M$ and $K$ are subgroups preserving $S$ by left-multiplication, then so is their join $<M, K>$.
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.)