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

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.


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 the previous exercise, 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.

Advertisements
Post a comment or leave a trackback: Trackback URL.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: