Question

If K and N are normal subgroups of a group G , prove that $K\cap N$ is a normal subgroup of G .

Short answer

It is proved that $K\cap N$ is a normal subgroup of G .

Step-by-step solution

Required Theorem

Theorem 8.11: The following conditions on a subgroup N of a group G are equivalent:

1. Nis a normal subgroup of G.
2. for every , Where
3. for every , Where
4. for every .
5. for every .

Proving that that K∩N is a normal subgroup of G

Let a be any arbitrary element in G; $a\in G$.

Now to prove that $K\cap N$ is a normal subgroup of G , we have to show that $a\left(K\cap N\right)a^{-1}\subset K\cap N$

Supposek and n are two elements, $k\in K,$and $n\in N$.

Both K and N are normal subsets. So,

$ak{a}^{-1}\in k$and $an{a}^{-1}\in N$.

Now considering , $g\in K\cap N$, $a\in g$

Since $K\cap N$ is an intersection on both K and N and both are normal subgroups. So for any element of $K\cap N$ , we can write

$ag{a}^{-1}\in K\cap N$.

This implies that$ag{a}^{-1}\subseteq K\cap N$ .

Hence, from the above result and theorem 8.11, we can conclude that $K\cap N$ is a normal subgroup of G .