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Identity component
In mathematics, the identity component of a topological group G is the connected component G0 of G that contains the identity element of the group. Similarly, the identity path component of a topological group G is the path component of G that contains the identity element of the group.
Properties
The identity component G0 of a topological group G is a closed normal subgroup of G. It is closed since components are always closed. It is a subgroup since multiplication and inversion in a topological group are continuous maps by definition. Moreover, for any continuous automorphism a of G we have
- a(G0) = G0.
Thus, G0 is a characteristic subgroup of G, so it is normal.
The identity component G0 of a topological group G need not be open in G. In fact, we may have G0 = {e}, in which case G is totally disconnected. However, the identity component of a locally path-connected space (for instance a Lie group) is always open, since it contains a path-connected neighbourhood of {e}; and therefore is a clopen set.
The identity path component may in general be smaller than the identity component (since path connectedness is a stronger condition than connectedness), but these agree if G is locally path-connected.
Component group
The quotient group G/G0 is called the group of components or component group of G. Its elements are just the connected components of G. The component group G/G0 is a discrete group if and only if G0 is open. If G is an affine algebraic group then G/G0 is actually a finite group.
One may similarly define the path component group as the group of path components (quotient of G by the identity path component), and in general the component group is a quotient of the path component group, but if G is locally path connected these groups agree. The path component group can also be characterized as the zeroth homotopy group, \( \pi_0(G,e). \)
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.
References
Lev Semenovich Pontryagin, Topological Groups, 1966.
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