I have the following question consisting of two parts: a) Let G be a finite group. Show that if H<G then
G is not the union of conjugates of H ?
b) Show that if G acts transitively on a set X of size at least 2
then some g in G acts without fixed points (Hint: Use a)
For a) I tried to use a counting argument.
Since |gHg^-1|=|H| then The total number of elements
are (|G|/|N(H)|).(|H|-1)<= (|G|/|H|).(|H|-1) <= ...
Does that help me in any thing ?
For b) I can say , let a,b in G then there is g s.t. g.a=b
and we can show that G_a=g^-1.G_b.g
from a) we can conclude there is an element g such that
is not included in any conjugate subgroup of H.
Can any one help on this question ? Thanks