Let H be a normal subgroup of a finite group G and let a be an elementof G. Complete the following statement: The order of the elementaH in the factor group G/H is the smallest positive integer nsuch that an is .

Prove that If |G|=np with p>n and p prime, and H is a subgroup of G with order p, then: H is normal in G.   Please be as clear as possible showing and explaining all the steps, and use definitions if necessary. Thank you very much.

Let G be a group and a ∈ G. A conjugate of a is any element of the form bab-1 for some b ∈ G. Show that if c = bab-1 is a conjugate of a in G and n is any integer, then cn=banb-1.  First prove for n ≥ 0 by induction on n.

Consider a finite group G with 60 elements. Let H be a subgroup of G with 12 elements. Prove that for any two elements a and b in G, if ab belongs to H, then ba also belongs to H.