Question

A mapstudent submitted image, transcription available below: S--->S' of G-sets is called a homomorphism of G-sets ifstudent submitted image, transcription available below(gs)=gstudent submitted image, transcription available below(s) for all s in S and g in G. Letstudent submitted image, transcription available belowbe such a homomorphism. Prove the following:

(a) The stabilizer Gstudent submitted image, transcription available below(s) contains the stabilizer Gs

Answer 1

To prove that the stabilizer in the homomorphism's target G_{S'(s')} contains the stabilizer in the source G_s, we show that if g fixes s in the source, the homomorphism implies it also fixes the image s' in the target, thus proving the containment.

Step-by-step explanation:

A map S → S' of G-sets is called a homomorphism of G-sets if (gs) = gS(s) for all s in S and g in G. To prove that the stabilizer G_{S'(s')} contains the stabilizer G_s for a homomorphism φ: S → S' with s' = φ(s), let's consider an element g in G_s. By definition of the stabilizer, g keeps s fixed, so gs = s. Applying the homomorphism, we'll have φ(gs) = φ(s), which implies gS'(s') = s'. This shows that g also fixes s', so g is in the stabilizer G_{S'(s')}. Thus, every element of G_s is also an element of G_{S'(s')}, proving that G_s ⊆ G_{S'(s')}.