Root group
Encyclopedia
A root group 
is a group
together with a set of prime number
s
satisfying the axiom:
.
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.
is a groupGroup (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
together with a set of prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s
satisfying the axiom:.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
Abelian root group
If G is an abelian group and P is a set of primes then G is an abelian P-root group if every element in G has a pth root for every prime p in P:g\in G,p\in P \Rightarrow \exists h\in G, h^p=g\;...
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 groupSpecial unitary groupThe special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...
s and special orthogonal groups are root groups for all primes
. For example, every element of 
except the identity is a rotation and has 
th roots. For 
the identity of 
has an infinite number of 
-roots for any prime 
, and the same is true of 
for 
.
- The orthogonal groupOrthogonal groupIn mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
s are
-root groups for 
the set of all odd primes, but are not 2-root groups.