Identity Component - Examples

Examples

  • The group of non-zero real numbers with multiplication (R*,•) has two components and the group of components is ({1,−1},•).
  • Consider the group of units U in the ring of split-complex numbers. In the ordinary topology of the plane {z = x + j y : x, y ∈ R}, U is divided into four components by the lines y = x and y = − x where z has no inverse. Then U0 = { z : |y| < x } . In this case the group of components of U is isomorphic to the Klein four-group.

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