Find ( N_G(H) ): Elements that normalize ( H ). By inspection, ( H ) is normalized by any permutation that permutes the three pairs ( 1,2, 3,4 ), etc. Actually, known fact: ( H ) is normal in ( S_4 ) but let's check: Conjugate ( (12)(34) ) by (12): ( (12)(12)(34)(12) = (12)(34) ) (same). Conjugate by (13): ( (13)(12)(34)(13) = (14)(23) \in H ). So indeed, all conjugates remain in ( H ). Thus ( N_G(H) = S_4 ).
Kernel: ( \ker \varphi = g \in G \mid g \cdot aH = aH \ \forall a \in G ). That means ( gaH = aH ) for all ( a ) (\Rightarrow) ( a^-1gaH = H ) for all ( a ) (\Rightarrow) ( a^-1ga \in H ) for all ( a ) (\Rightarrow) ( g \in \bigcap_a \in G aHa^-1 = \textcore_G(H) ).
: This exercise is standard in any "Dummit Foote solutions Chapter 4" PDF. Understand this proof thoroughly—it reapplies in Sylow theory. Exercise 4.4.8: Action on Cosets Problem : Let ( H \le G ) with index ( n ). Prove there exists a homomorphism ( \varphi: G \to S_n ) with kernel contained in ( H ).
: This is the foundation for the proof of Cayley’s theorem and the existence of normal subgroups of small index. Exercise 4.5.4: Conjugation on Subgroups Problem : Let ( G = S_4 ). Find the orbit and stabilizer of the subgroup ( H = e, (12)(34), (13)(24), (14)(23) ) under conjugation.
