Suppose that a group G has a subgroup of order n Prove that

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.

Solution

Let H[i] denote the subgroups of order n, where i ranges over any index set.

Let H = H[i], the intersection of them all.

Now examine the expression
aHa¹ = a ( H[i] ) a¹ = aH[i]a¹

now each aH[i]a¹ is a subgroup of order n too... so the operation sending H[i] to aH[i]a¹ just rearranges the subgroups. In particular, the intersection of these subgroups is still the same, since we\'ve done nothing but rearranged the order of the intersection:

aHa¹ = a ( H[i] ) a¹ = aH[i]a¹ = H[j] = H

Thus H is normal.