Question

A copy of Klein 4-group K in S4 is V = {(1),(12)(34),(13)(24),(14)(23)}. Use H = {(1),(12)(34)} to prove that in general H ◅ K ◅ G ◅ G ⇒ H ◅ G
Because in this case H ◅ V and V ◅ S4, H ◅ S4
A) Assume V ≤ S4 and prove that V ◅ S4. You can use Theorem .6, for instance listing and comparing left and right cosets of V in S4
B) Assume H ≤ V and prove that H ◅ V. You can use theorem .6, for instance listing and comparing left and right cosets of H in V.
C) Assume H ≤ S4 and prove that H ◅ S4. You can use theorem .6, for instance listing and comparing left and right cosets of H in S4

Answer 1

the ⇒ in H ◅ K ◅ G ◅ G ⇒ H ◅ G should be marked through