A criterion for the equality of a left and right coset

Let G be a group, g \in G, and N \leq G. Prove that gN = Ng if and only if g \in N_G(N).


(\Rightarrow) Suppose gN = Ng. Right multiplying by g^{-1}, we have gNg^{-1} = Ngg^{-1} = N, so that g \in N_G(N). (\Leftarrow) Suppose g \in N_G(N). Then gNg^{-1} = N, and right multiplying by g we have gN = Ng.

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: