Does a Factor Group Element's Order Imply the Same Order in the Original Group?

[PsychonautQQ]

Homework Statement

Let G be a finite group and let K be normal to G. If the factor group G/K has an element of order n, show that G has an element of order n.

Homework Equations

None

The Attempt at a Solution

Lets say Kg is the element in G/K with order n.
That means:
(Kg)^n = K

and from properties of factor groups we know:
(Kg)^n = Kg^n
so Kg^n = K
hence g^n = 1
if g^n = 1 then it must be in G, because G has the identity (1).

Is this correct thinking?

 

[Dick]

Homework Statement

Let G be a finite group and let K be normal to G. If the factor group G/K has an element of order n, show that G has an element of order n.

Homework Equations

None

The Attempt at a Solution

Lets say Kg is the element in G/K with order n.
That means:
(Kg)^n = K

and from properties of factor groups we know:
(Kg)^n = Kg^n
so Kg^n = K
hence g^n = 1
if g^n = 1 then it must be in G, because G has the identity (1).

Is this correct thinking?

No. Haven't you posted this before? Kg^n=K only tells you that g^n is in K.