Question

Show that if N and K are normal subgroups of a group G, then N∩K is a normal subgroup of G.

Answer 1

To show that the intersection of two normal subgroups, N and K, is also a normal subgroup of a group G, we need to prove two conditions: closure under the group operation and containment of conjugate elements.

Step-by-step explanation:

To show that the intersection of two normal subgroups, N and K, is also a normal subgroup of a group G, we need to prove two conditions:

N∪K is a subgroup of G.

For any g in G, g(N∪K)g-1 is contained in N∪K.

To prove the first condition, let's consider two elements a and b that are in N∪K. Since N and K are normal subgroups of G, a and b are in N and K respectively. Therefore, a and b are also in N∪K, showing closure under the group operation.

Now, to prove the second condition, let's take any element g in G. We need to show that g(N∪K)g-1 is contained in N∪K. Since N and K are normal subgroups, we have gNg-1 is contained in N and gKg-1 is contained in K. Therefore, gNg-1 and gKg-1 are both contained in N and K respectively. Hence, their intersection gNg-1 ∩ gKg-1 is contained in N∪K.