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 $x \in G, |x| = |\varphi (x)|$.