Commuting probability
   HOME

TheInfoList



OR:

In mathematics and more precisely in group theory, the commuting probability (also called degree of commutativity or commutativity degree) of a
finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...
is the probability that two randomly chosen elements
commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...
. It can be used to measure how close to
abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...
a finite group is. It can be generalized to infinite groups equipped with a suitable
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more gener ...
, and can also be generalized to other
algebraic structures In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
such as rings.


Definition

Let G be a
finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...
. We define p(G) as the averaged number of pairs of elements of G which commute: :p(G) := \frac \#\left\. If one considers the
uniform distribution Uniform distribution may refer to: * Continuous uniform distribution * Discrete uniform distribution * Uniform distribution (ecology) * Equidistributed sequence In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be ...
on G^2, p(G) is the probability that two randomly chosen elements of G commute. That is why p(G) is called the commuting probability of G.


Results

* The finite group G is abelian if and only if p(G) =1. * One has ::p(G) = \frac : where k(G) is the number of conjugacy classes of G. * If G is not abelian, then p(G) \leq 5/8 (this result is sometimes called the 5/8 theorem) and this upper bound is sharp: there is an infinity of finite groups G such that p(G) = 5/8, the smallest one is the
dihedral group of order 8 Some elementary examples of groups in mathematics are given on Group (mathematics). Further examples are listed here. Permutations of a set of three elements Consider three colored blocks (red, green, and blue), initially placed in the order R ...
. * There is no uniform lower bound on p(G). In fact, for every positive integer n, there exists a finite group G such that p(G) =1/n. * If G is not abelian but simple, then p(G) \leq 1/12 (this upper bound is attained by \mathfrak_5, the alternating group of degree 5). * The set of commuting probabilities of finite groups is reverse-well-ordered, and the reverse of its order type is known to be either \omega^\omega or \omega^.


Generalizations

* The commuting probability can be defined for others
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
s such as finite rings. * The commuting probability can be defined for infinite compact groups; the probability measure is then, after a renormalisation, the
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though ...
.


References

{{Reflist Finite groups