If you find any mistakes, please make a comment! Thank you.

A group action is faithful precisely when the kernel of the corresponding permutation representation is trivial


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.6

Prove that a group $G$ acts faithfully on a set $A$ if and only if the kernel of the action is the set containing only the identity.


Solution: We know that a group action is faithful precisely when the corresponding permutation representation $\varphi : G \rightarrow S_A$ is injective. Moreover, a group homomorphism is injective precisely when its kernel is trivial. Finally, by Exercise 1.7.5, the kernel of a group action is equal to the kernel of the corresponding permutation representation. So $G$ acts faithfully on $A$ if and only if the kernel of the action is trivial.

Linearity

This website is supposed to help you study Linear Algebras. Please only read these solutions after thinking about the problems carefully. Do not just copy these solutions.

Leave a Reply

Close Menu