Properties
- The kernel of an isomorphism from (G, *) to (H, ), is always {eG} where eG is the identity of the group (G, *)
- If (G, *) is isomorphic to (H,), and if G is abelian then so is H.
- If (G, *) is a group that is isomorphic to (H, ), then if a belongs to G and has order n, then so does f(a).
- If (G, *) is a locally finite group that is isomorphic to (H, ), then (H, ) is also locally finite.
- The previous examples illustrate that 'group properties' are always preserved by isomorphisms.
Read more about this topic: Group Isomorphism
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)