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)|.
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