Question

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

Short answer

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

1. Nis a normal subgroup of G.
2. $a^{-1}Na\subseteq N$for every$a\in G$,Where $a^{-1}Na\subseteq N$$\left\{a^{-1}Na| n\in N\right\}$
3. $aN{a}^{-1}\subseteq N$for every $a\in G$ , Where $aN{a}^{-1}\subseteq N$$\left\{aN{a}^{-1}| n\in N\right\}$
4. $a^{-1}Na\subseteq N$for every $a\in G$ .
5. $aN{a}^{-1}\subseteq N$for every $a\in G$ .

Step-by-step solution

Proving that K∩N is a normal subgroup of K

Suppose $n\in N\cap K,$ and $a\in k$

As N is a normal subgroup ofG, we can say that:

$an{a}^{-1}\in N$

Also, because K is a subgroup of G, we get

$an{a}^{-1}\in K$

Therefore, $a\left(N\cap K\right)a^{-1}\in N\cap K$ .

Thus $a\left(N\cap K\right)a^{-1}\subseteq N\cap K$ for $a\in K$.

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