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:
“There is not much that even the most socially responsible scientists can do as individuals, or even as a group, about the social consequences of their activities.”
—Eric J. Hobsbawm (b. 1917)
“The horror of Gandhi’s murder lies not in the political motives behind it or in its consequences for Indian policy or for the future of non-violence; the horror lies simply in the fact that any man could look into the face of this extraordinary person and deliberately pull a trigger.”
—Mary McCarthy (1912–1989)
“War is thus divine in itself, since it is a law of the world. War is divine through its consequences of a supernatural nature which are as much general as particular.... War is divine in the mysterious glory that surrounds it and in the no less inexplicable attraction that draws us to it.... War is divine by the manner in which it breaks out.”
—Joseph De Maistre (1753–1821)