Exercises for the Course Math

(1) Let G be a group and H ⊂ Z(G) a subgroup such that G/H is cyclic. Prove that G is abelian. Conclude that every group of order p2 (p a prime number) is abelian and that the centre of any non-abelian group of order p3 has p elements. (2) (?) Let H be a normal subgroup of a p-group G, H 6= {e}. Prove that H∩Z(G) 6= {e}. In particular, one obtains that any normal subgroup with p-elements is contained in Z(G). (3) Let G be a p-group and H < G a proper subgroup with pk elements. Prove that there is a subgroup of G with pk+1 elements that contains H. Deduce that every maximal subgroup of a p group has index p. (4) (?) Let G be a finite group and H a normal subgroup of G. Let P be a Sylow subgroup of G. Prove that H ∩ P is a Sylow subgroup of H and HP/H is a Sylow subgroup of G/H. (5) Show that up to isomorphism there are precisely 5 groups of order 8 given by Z/8Z,Z/4Z× Z/2Z, (Z/2Z)3, D8 and Q. D8 is the dihedral group with 8 elements and Q is the quaternion group {±1,±i,±j,±k} with i2 = j2 = k2 = −1, ij = −ji = k (±1 commute with any element). (6) Prove the Cauchy-Frobenius formula (also known as Burnside’s lemma). Let G be a finite group acting on a finite non-empty set S. Let N be the number of orbits of G in S. Then