Consequences
From the definition, it follows that any isomorphism will map the identity element of G to the identity element of H,
that it will map inverses to inverses,
and more generally, nth powers to nth powers,
for all u in G, and that the inverse map is also a group isomorphism.
The relation "being isomorphic" satisfies all the axioms of an equivalence relation. If f is an isomorphism between two groups G and H, then everything that is true about G that is only related to the group structure can be translated via f into a true ditto statement about H, and vice versa.
Read more about this topic: Group Isomorphism
Famous quotes containing the word consequences:
“The consequences of our actions grab us by the scruff of our necks, quite indifferent to our claim that we have “gotten better” in the meantime.”
—Friedrich Nietzsche (1844–1900)
“If you are prepared to accept the consequences of your dreams ... then you must still regard America today with the same naive enthusiasm as the generations that discovered the New World.”
—Jean Baudrillard (b. 1929)
“The medium is the message. This is merely to say that the personal and social consequences of any medium—that is, of any extension of ourselves—result from the new scale that is introduced into our affairs by each extension of ourselves, or by any new technology.”
—Marshall McLuhan (1911–1980)