# Prove that the intersection of all subgroups of G of order n is a normal subgroup of

1. Apr 9, 2010

### tyrannosaurus

1. The problem statement, all variables and given/known data
Suppose that a group G has a subgroup of order n. Prove that the intersection of all subgroups of G of order n is a normal subgroup of G.

2. Relevant equations

3. The attempt at a solution
I know that I need to do the following:
Let S be the set of all subgroups of G with order N and let C be the set of all conjugates of those subgroups.

(1) Prove the conjugate of the intersection of all elements in the set S is contained in the intersection of all the elements of C. [I.e. if K is the intersection of all H in S, then gK(g^-1) is contained in the intersection of all gH(g^-1)]

(2) Prove that the sets S and C are the same (i.e. S=C) by double inclusion and conclude then that the intersection of all the elements of S must be the same as the intersection of all the elements of C.

(3) Finally, use one of the normality tests to conclude that the intersection of all the elements of S is normal in G.
However, I do not know how to set up any of these proofs.

2. Apr 10, 2010

### eok20

Re: Prove that the intersection of all subgroups of G of order n is a normal subgroup

I think you can do this rather directly: let g be in the intersection of all subgroups of order n and let h be in G. Then you need to show that hgh^(-1) is in all subgroups of order n. So let K be any subgroup of order n and try to show hgh^(-1) is in it. If we conjugate a group of order n then we get another group of order n. So g is in every conjugate of K. What conjugate of K should we choose to help us finish the proof?