A criterion for normalcy of finite subgroups

Let G be a group and N a finite subgroup of G. Show that gNg^{-1} \subseteq N if and only if gNg^{-1}. Deduce that N_G(N) = \{ g \in G \ |\ gNg^{-1} \subseteq N \}.


It suffices to show that for all g \in G, gNg^{-1} \subseteq N implies gNg^{-1} = N. Let g \in G. The mapping n \mapsto gng^{-1} is a bijection N \rightarrow gNg^{-1}, so that |gNg^{-1}| = |N|. Since N is finite, gNg^{-1} = N.

The last statement follows trivially.

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: