Question

Prove that Inn $G$ is a normal subgroup of Aut$G$ . [See Exercise 37 of Section 7.4.]

Short answer

It has been proved thatInn $G$ is a normal subgroup of Aut$G$.

Step-by-step solution

Define Inn  G

It is given that in a group $G$ , Inn $G$ is the set of all inner automorphism of $G$ , that is, isomorphisms of the form $f\left(a\right)=c^{-1}ac$ for some $c\in G$.

Prove gfc∈InnG

Let $g\left(c\right)$ be the inner automorphism induced by $c$ for $c\in G$:$g\left(c\right)\left(x\right)=cx{c}^{-1}$ .

If $f\in Aut\left(G\right)$, then

$\begin{array}{c}\left(f\circ g\left(c\right)\right)\left(x\right)=f\left(g\left(c\right)\left(x\right)\right)\\ =f\left(cx{c}^{-1}\right)\\ =f\left(c\right)f\left(x\right)f\left(c^{-1}\right)\\ =g\left(f\left(c\right)\right)\left(f\left(x\right)\right)\\ =\left(g\left(f\left(c\right)\right)\circ f\right)\left(x\right)\end{array}$

Therefore, $f\circ g\left(c\right)=g\left(f\left(c\right)\right)\circ f$ so that $f\circ g\left(c\right)\circ f^{-1}=g\left(f\left(c\right)\right)\in InnG$ .

Conclusion

Thus it can be concluded that Inn $G$ is normal in $AutG$ .