Stabiliser (of a group action)

Let the group \(G\) act on the set \(X\). Then for each el­e­ment \(x \in X\), the sta­bil­iser of \(x\) un­der \(G\) is \(\mathrm{Stab}_G(x) = \{ g \in G: g(x) = x \}\). That is, it is the col­lec­tion of el­e­ments of \(G\) which do not move \(x\) un­der the ac­tion.

The sta­bil­iser of \(x\) is a sub­group of \(G\), for any \(x \in X\). (Proof.)

A closely re­lated no­tion is that of the or­bit of \(x\), and the very im­por­tant Or­bit-Sta­bil­iser the­o­rem link­ing the two.

Parents:

  • Group action

    “Groups, as men, will be known by their ac­tions.”