Editing Compact group (section)

===Fundamental group and center===
{{See also|Fundamental group#Lie groups}}
It is important to know whether a connected compact Lie group is simply connected, and if not, to determine its [[fundamental group]]. For compact Lie groups, there are [[Fundamental group#Lie groups|two basic approaches]] to computing the fundamental group. The first approach applies to the classical compact groups <math>\operatorname{SU}(n)</math>, <math>\operatorname{U}(n)</math>, <math>\operatorname{SO}(n)</math>, and <math>\operatorname{Sp}(n)</math> and proceeds by induction on <math>n</math>. The second approach uses the root system and applies to all connected compact Lie groups.

It is also important to know the center of a connected compact Lie group. The center of a classical group <math>G</math> can easily be computed "by hand," and in most cases consists simply of whatever roots of the identity are in <math>G</math>. (The group SO(2) is an exception—the center is the whole group, even though most elements are not roots of the identity.) Thus, for example, the center of <math>\operatorname{SU}(n)</math> consists of ''n''th roots of unity times the identity, a cyclic group of order <math>n</math>.

In general, the center can be expressed in terms of the root lattice and the kernel of the exponential map for the maximal torus.<ref>{{harvnb|Hall|2015}} Section 13.8</ref> The general method shows, for example, that the simply connected compact group corresponding to the exceptional root system <math>G_2</math> has trivial center. Thus, [[G2 (mathematics)|the compact <math>G_2</math> group]] is one of very few simple compact groups that are simultaneously simply connected and center free. (The others are [[F4 (mathematics)|<math>F_4</math>]] and [[E8 (mathematics)|<math>E_8</math>]].)