Question

If $M$is a characteristic subgroup of $N$and $N$ is a normal subgroup of a group$G$ , prove that $M$ is a normal subgroup of $G$ . [See Exercise 11.]

Short answer

It has been proved that$M$ is a normal subgroup of $G$.

Step-by-step solution

Given that

It is given that $N$ is a normal subgroup of a group $G$.

$M$is a characteristic subgroup of $N$.

We know from the hint that a subgroup $N$ of a group $G$ is said to be characteristic if $f\left(N\right)\subseteq N$ for every automorphism $f$ of $G$.

Let $g\left(c\right)$ be the inner automorphism induced by$c$ .

Prove that  M is a normal subgroup of  G

Since $N$ is normal, then $g\left(c\right)=cN{c}^{-1}=N$ .

Thus, the restriction of $g\left(c\right)$to $N$gives an automorphism$\alpha \in AutN$ .

Since $M$ is a characteristic in$N$ , by definition, it is clear that $\alpha \left(M\right)\subseteq M$.

This implies that $cM{c}^{-1}\subseteq M$ for any arbitrary $c\in G$.

Conclusion

Thus, it can be concluded that $M$ is a normal subgroup of $G$.