i worked out part c,
still trying find a solid process of showing that F is the smallest normal subgroup containing H
hey,
i'm trying to study for my up coming exams with these past papers but i've gotten stuck with part c,
i've tried a bunch of different spanning tree's some gave better generators and some gave worse but the ones in the picture were the first ones i found,
i've been trying to use the generators to show that the its not possible to get the conjugate of one of the provided elements of H by multiplying the generators but it hasn't gotten me anywhere,
i also have a feeling the answer to part d) is the whole group F but im not sure why it's just a guess.
could anyone give me a hand working out these two parts?
thanks in advance
let K be the smallest normal subgroup of F that contains H.
H partitions F into 5 cosets, so H must partition K into at most 5 cosets (two elements x,y of F are in the same coset of H even when both are in K). thus [K:H] is finite. similarly, [F:K] must also be finite (each coset of H is entirely contained in a coset of K).
hence [F:H] = [F:K][K:H]. since [F:H] = 5 is prime, either [F:K] = 1, or [K:H] = 1. but [K:H] = 1 is not possible, since that implies H = K, but H is not normal.
thus [F:K] = 1, which means F = K.