Does 'G leaves W stable' mean GW=W or GW⊂W?

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

jostpuur said:
If [itex]G\subset \textrm{End}(V)[/itex], and [itex]W\subset V[/itex] is a subspace of a vector space V, and somebody says "G leaves W stable", does it mean [itex]GW=W[/itex] or [itex]GW\subset W[/itex] 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 [itex]A \subset X[/itex], some [itex]g \in G[/itex] takes [itex]A \mapsto B \subset A[/itex]. 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.)