- 1 Up To Isomorphism There Are Five Transitive Subgroups Of S4 They Are The Cyclic Group C4 The Klein Four Group C 1 (158.17 KiB) Viewed 31 times
1. Up to isomorphism, there are five transitive subgroups of S4. They are the cyclic group C4, the Klein four group C₂ × C2, the dihedral group D4, the alternating group A and the symmetric group S4. (a) Convince yourself you are aware of these groups. = (b) Let E/F be a degree four separable field extension. Is it always the case that there exists a field F≤G ≤ E with [G: F] = 2? [Hint: Let K be the Galois closure of E (so if E is given by adjoining a root of a quartic polynomial f(x), then K is the splitting field of F). Look at the list of five possibilities for Gal(K/F), each of which can occur. (Please assume that each of these can occur in order to solve this question, or prove it!)]