In mathematics and more precisely in
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
, 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 ma ...
is the
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
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 ge ...
,
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 of ...
such as
rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
.
Definition
Let
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 ma ...
. We define
as the averaged number of pairs of elements of
which commute:
:
If one considers the
uniform distribution
Uniform distribution may refer to:
* Continuous uniform distribution
* Discrete uniform distribution
* Uniform distribution (ecology)
* Equidistributed sequence
See also
*
* Homogeneous distribution
In mathematics, a homogeneous distribution ...
on
,
is the probability that two randomly chosen elements of
commute. That is why
is called the commuting probability of
.
Results
* The finite group
is abelian if and only if
.
* One has
::
: where
is the number of
conjugacy class
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wo ...
es of
.
* If
is not abelian, then
(this result is sometimes called the 5/8 theorem) and this upper bound is sharp: there is an infinity of finite groups
such that
, the smallest one is the
dihedral group of order 8.
* There is no uniform lower bound on
. In fact, for every positive integer
, there exists a finite group
such that
.
* If
is not abelian but
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
, then
(this upper bound is attained by
, the
alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or
Basic pr ...
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
or
.
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 o ...
s such as
finite ring
In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements.
Every finite field is an example of a finite ring, and the additive part of every finite ring is an example of an abelian finite grou ...
s.
* The commuting probability can be defined for infinite
compact group
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
s; 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, thou ...
.
References
{{Reflist
Finite groups