## Related questions with answers

Question

Prove that $G^{i}$ is a characteristic subgroup of G for all i.

Solution

VerifiedStep 1

1 of 2Let $\phi\in Aut(G)$ and $a,b\in G$. Then

$\phi([a,b])=\phi(a^{-1}b^{-1}ab)=\phi(a)^{-1}\phi(b)^{-1}\phi(a)\phi(b)=[\phi(a),\phi(b)]$

so $\phi([a,b])\in [G,G]$ and therefore $G^1$ is a characteristic subgroup. Assume now that for $i>1$ we have that $G^k$ is a characteristic subgroup of $G$ for all $k<i$. Then for $\phi\in Aut(G)$, $a\in G$ and $b\in G^{i-1}$ we have again that

$\phi([a,b])=\phi(a^{-1}b^{-1}ab)=\phi(a)^{-1}\phi(b)^{-1}\phi(a)\phi(b)=[\phi(a),\phi(b)]$

so by the induction hypothesis $\phi([a,b])\in G^i$ and therefore $G^i$ is a characteristic subgroup.

## Create a free account to view solutions

By signing up, you accept Quizlet's Terms of Service and Privacy Policy

## Create a free account to view solutions

By signing up, you accept Quizlet's Terms of Service and Privacy Policy

## Recommended textbook solutions

## More related questions

1/4

1/7