. 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 can easily be computed "by hand," and in most cases consists simply of whatever roots of the identity are in . (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 consists of nth roots of unity times the identity, a cyclic group of order . In general, the center can be expressed in terms of the root lattice and the kernel of the exponential map for the maximal torus. The general method shows, for example, that the simply connected compact group corresponding to the exceptional root system has trivial center. Thus, the compact group is one of very few simple compact groups that are simultaneously simply connected and center free. (The others are and .) Further examples Amongst groups that are not Lie groups, and so do not carry the structure of a manifold, examples are the additive group Zp of p-adic integers, and constructions from it. In fact any profinite group is a compact group. This means that Galois groups are compact groups, a basic fact for the theory of algebraic extensions in the case of infinite degree. Pontryagin duality provides a large supply of examples of compact commutative groups. These are in duality with abelian discrete groups. Haar measure Compact groups all carry a Haar measure,<ref></ref> which will be invariant by both left and right translation (the modulus function must be a continuous homomorphism to positive reals (R+, ×), and so 1). In other words, these groups are unimodular. Haar measure is easily normalized to be a probability measure, analogous to dθ/2π on the circle. Such a Haar measure is in many cases easy to compute; for example for orthogonal groups it was known to Adolf Hurwitz, and in the Lie group cases can always be given by an invariant differential form. In the profinite case there are many subgroups of finite index, and Haar measure of a coset will be the reciprocal of the index. Therefore, integrals are often computable quite directly, a fact applied constantly in number theory. If is a compact group and is the associated Haar measure, the Peter–Weyl theorem