Foote Solutions Chapter 4 - Dummit

: 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.

Thus ( p^2 = |Z(G)| + kp ), where ( k ) = number of non-central conjugacy classes. dummit foote solutions chapter 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 is the foundation for the proof

This kernel is a normal subgroup of ( G ) contained in ( H ). . Thus ( p^2 = |Z(G)| + kp ),