Today I Learned

Some of the things I've learned every day since Oct 10, 2016

106: Group Isomorphisms

In group theory, a group isomorphism between 2 groups G, H does not simply refer to an ‘isomorphism’ between the underlying sets of G, H, but to a special type of homomorphism between them: one which is bijective. In a sense, isomorphic groups are the same group, just with different symbols representing their elements and operations — something which is not necessarily true if they’re homomorphic or if just their underlying sets are isomorphic.

In particular, if \varphi : G \mapsto H is a group isomorphism, then

  • |G| = |H|
  • G is abelian \leftrightarrow H is abelian
  • for all x \in G, |x| = |\varphi (x)|.

Leave a Reply

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

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

Google+ photo

You are commenting using your Google+ 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 )


Connecting to %s

%d bloggers like this: