Component groups

Last updated: October 8, 2005

Let G be a connected complex reductive algebraic group, with a fixed inner class, and fix a real form in this inner class, represented by an involution θ. Let G(R) be the group of real points of G for the real form, and denote K the fixed point group of θ. As explained here, K is (isomorphic to) the complexification of a maximal compact subgroup of G(R); in particular, the component group of K is equal to the component group of G(R).

The following facts are known:

From these facts, it follows immediately that the component group of G(R) is always an elementary abelian 2-group (because this is true for tori), and in fact it is possible to deduce an algorithm to compute the group from the root datum of G and the involution θ. The algorithm is explained in the 2004 notes by Fokko du Cloux here. This information is available in the program through the "components" command.
[1] R. Steinberg, Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc. 80 (1968), pp. 1-108.

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