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**hgd7833** 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