Assignment:
Q1.) If |G|=p^n for some prime p, and 1 is not equal to H, a normal subgroup of G, show that H intersect Z(G) is not equal to 1.
Q2.) Suppose G is finite, H is a subgroup of G, [G:H]=n and |G| does not divide n!. Show that there is a normal subgroup K of G, with K not equal to 1, such that K is a subgroup of H.
Q3.) Prove that if G contains no subgroup of index 2, then any subgroup of index 3 is normal in G.
Q4.) Suppose G acts on S, x in G, and x in S. Show that Stab_G(xs)=xStab_G(s)x^-1.
Q5.) Let G be a group of order 15, which acts on a set S with 7 elements Show the group action has a fixed point.
Q6.)a.) Suppose that H and K both have finite index in G. Prove that [G:H intersect K]is a subgroup of [G:H][G:K].
b.) Suppose that [G:H] is finite. Prove that H contains a normal subgroup of G which, in turn, has finite index in G.
Provide complete and step by step solution for the question and show calculations and use formulas.