Group Isomorphism - Properties

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.

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Famous quotes containing the word properties:

    “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)

    “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)