Let N be a normal subgroup of a group G a If G is a cyclic g

Let N be a normal subgroup of a group G. (a) If G is a cyclic group, prove that G/N is cyclic. (b) If G is Abelian, prove that G/N is Abelian. (c) If g has order 25 in G, what are the possible orders of the element gN in G/N?

Solution

1) Let G be cyclic and N be (normal as G is abelian ) a subgroup of G.

Let x generate G.

Claim: x+N generates G/N

Proof: Any element of G/N is of the form y + N. But y = kx for some integer k (as x generates G)

it follows that y+N = kx + N = k(x+N) .

Hence the result

(2) If G is abelian, then G/N is abelian.

Proof: xN yN = xyN (by definition of quotient group)

                      =yx N (as G is abelian)

                        =yNxN

Thus any two cosets commute. So G/N is abelian

(3) If g has order 25 in G , what are the possible orders of gN in G/N?

Now the order of G/N divides the order of G (as the number of elements in G/N is the index [G:N], which divides the order of G.

The order of gN must divide the order of G/N, hence must divide 25.

Possible orders of gN are 1,5,25