What Happens When a Group Leaves a Subset of a Countably Infinite Set Stable?

[jostpuur]

If $G\subset \textrm{End}(V)$, and $W\subset V$ is a subspace of a vector space V, and somebody says "G leaves W stable", does it mean $GW=W$ or $GW\subset W$ or something else?

 

[matt grime]

It means it maps W to itself.

 

[Chris Hillman]

Finite dimensions

If $G\subset \textrm{End}(V)$, and $W\subset V$ is a subspace of a vector space V, and somebody says "G leaves W stable", does it mean $GW=W$ or $GW\subset W$ or something else?

Exercise: what can you say about these alternatives if V is finite dimensional?

If you read the "What is Information Theory?" thread: Exercise: suppose we have some infinite group G acting on some countably infinite set X. Suppose that for some $A \subset X$, some $g \in G$ takes $A \mapsto B \subset A$. What is the corresponding statement about stabilizers? If you have read Stan Wagon, The Banach-Tarski Paradox, what does this remind you of? Can you now give a concrete example (with illustration) of this phenomenon? (Hint: hyperbolic plane.)