i understand parts (a) and (b) I'm trying to prove part (c)
This is what i have in mind so far. 
am i on the right track or am i wrong? Please help.
2026-03-25 18:52:30.1774464750
Abstract Algebra. Stabilizers
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Your write ups could do with some formalizing and cleaning up; you are on the right track for your second inclusion. For the first inclusion, we take some permutation $\tau$ for which $\tau(TOP)=HAT$, and we must decompose it as $$ \tau=\sigma\rho,\rho\in Stab(TOP). $$ To this hand, note that since $\sigma(TOP)=HAT$, we must have $\sigma^{-1}(HAT)=TOP$. Thus $$ \sigma^{-1}\tau(TOP)=\sigma^{-1}(HAT)=TOP, $$ so $\sigma^{-1}\tau\in Stab(TOP)$. Can you finish from here?