A finite group of composite order n having a subgroup of every order dividing n is not simple

Let G be a finite group of composite order n with the property that for all k|n, G has a subgroup of order k. Prove that G is not simple.


Let p be the smallest prime dividing n, and write n = pm. Now G has a subgroup H of order m, and H has index p. By Corollary 5 in the text, H is normal in G.

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: