Root group

A root group G  is a group together with a set of prime numbers P = {p₁, p₂, ...}  satisfying the axiom:

g ∈ G ∧ p ∈ P ⇒ ∃h ∈ G, h^p = g .

To specify the set of primes, a group may be referred to as a P-root group. For a single prime p it may be referred to as a p-root group.

An abelian root group is such a group where the multiplication is commutative.

P-root groups may be further classified depending on whether the unit element has a non-trivial root for any or all of the primes in the set P.

Examples

• Every finite group with order coprime to all of the primes in the set P, or more generally any group such that the order of each element is coprime to all the primes in P is a P-root group.

• The special unitary groups and special orthogonal groups are root groups for all primes p . For example, every element of SO(3)  except the identity is a rotation and has p  p th roots. For n > 2  the identity of SO(n)  has an infinite number of p -roots for any prime p , and the same is true of SU(n)  for n > 1 .

• The orthogonal groups are P -root groups for P  the set of all odd primes, but are not 2-root groups.