Normalizer of of a sylow p-subgroup of G

Let Sp be a sylow P-subgroup G. Prove that the normalizer of the normalizer of Sp satisfied N(N(Sp))=N(Sp).

Prove that a group of order 120 is not simple.

Prove that a group of order 120 is not simple.

Homomorphism of cyclic group

Let G be a group and h:G-->h(G) a homomorphism.Prove or Disprove If G is cyclic,then h(G)is cyclic.

Automorphism of Zp+Zp ,p prime

let G=Zp+Zp .how many automorphism does G have. Please explain clearly the counting principle.

Groups and Generators

Let G =(x,y|x²ⁿ = e, xⁿ= y², yˉ¹ *x*y= x^ˉ¹). Show that Z(G) = {xⁿ , e} . Assume |G| = 4n, show that G/Z(G) ≈ Dn