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identity component

Contents

Definition

If GG is a topological group, then the identity component is the connected component of the identity element ee in GG.

Basic results

Proposition

The identity component G 0G_0 is a closed normal subgroup of GG.

Proof

It is clearly closed (indeed, any connected component is closed). If g,hG 0g, h \in G_0, then ghg h is in the same connected component as gg (since hh is in the same connected component as ee and left multiplication by gg is a homeomorphism), which in turn is in the same connected component as ee. Using similar reasoning, if gg is in the connected component as ee, then ee is in the same connected component as g 1g^{-1}. Hence G 0G_0 is a subgroup.

If ϕ\phi is any automorphism of GG, then ϕ(G 0)=G 0\phi(G_0) = G_0. (Indeed, ϕ(G 0)\phi(G_0) is a connected set containing ee and therefore ϕ(G 0)G 0\phi(G_0) \subseteq G_0. Replacing ϕ\phi by its inverse ϕ 1\phi^{-1}, we similarly have ϕ 1(G 0)G 0\phi^{-1}(G_0) \subseteq G_0 and therefore G 0ϕ(G 0)G_0 \subseteq \phi(G_0).) Applying this to inner automorphisms ϕ\phi, we conclude that G 0G_0 is a normal subgroup of GG.

  • Remark: G 0G_0 need not be open in GG; for example, for the group of pp-adic integers, G 0G_0 is the (non-open) singleton {e}\{e\}. However, if GG is locally connected, for example if GG is a Lie group, then G 0G_0 is open (and therefore also clopen. In this case, G/G 0G/G_0 is discrete (because GG/G 0G \to G/G_0 is an open map, implying that the identity and therefore every point in G/G 0G/G_0 is open).
Proposition

The group G/G 0G/G_0, equipped with the quotient space topology, is a Hausdorff topological group.

Proof

Given the fact that p:GG/G 0p \colon G \to G/G_0 is an open surjection, the product p×p:G×GG/G 0×G/G 0p \times p \colon G \times G \to G/G_0 \times G/G_0 is also an open surjection and therefore a quotient map. It follows easily from the universal property of quotient maps that the multiplication G×GGG \times G \to G therefore descends to a continuous multiplication G/G 0×G/G 0G/G 0G/G_0 \times G/G_0 \to G/G_0, so that G/G 0G/G_0 is a topological group.

Because a topological group is a uniform space, the Hausdorff condition follows from a weaker separation axiom such as T 1T_1 (points are closed). It suffices that the identity of G/G 0G/G_0 be closed. Its complement CC is the image under pp of the complement of G 0G_0 in GG (just by examining coset decompositions), which is open. Since pp is an open map, it follows that CC is open, so that {e}\{e\} is closed, as desired.

Created on February 3, 2012 23:46:11 by Todd Trimble (74.88.146.52)