If is a finite group, we define its commutativity degree as the probability that two elements of commute, that is
It is surprising that the value of may give strong information about . In particular, we prove here the two following results:
Theorem 1: Let be a finite group. If then is abelian.
Theorem 2: Let be a finite group. If then is nilpotent.
In fact, the two theorems above will follow from the upper degree equation bound:
Theorem 3: Let be a finite group and let denote its commutator subgroup. Then
Indeed, if is non-abelian then , and the upper degree equation bound implies ; therefore, theorem 1 is proved. Then, if then the upper degree equation bound implies . If , is abelian and there is nothing to prove. Otherwise, ie. for some . For all , , so either or which implies ; therefore, . We proved that , so ; in particular is nilpotent.
In order to prove Theorem 3, we first notice the following link between commutativity degree and the number of conjugacy classes of :
Lemma 1: Let be a finite group. Then where denotes the number of conjugacy classes of .
Proof. If , let denote the centralizer of and denote the conjugacy classes of ; for convenience, let be an element of . Then
Finally, the lemma follows.
Therefore, computing is equivalent to computing and . But we know that the number of conjugacy classes is quite related to representation theory, and indeed we will prove Theorem 3 thanks to representation theory.
Definitions: Let be a finite group. A (complex) representation is a morphism for some complex vector space (of finite dimension); the degree of the representation is the dimension of . By extension, we often say that is a representation.
Any finite group admits a representation. For example, if denotes the complex vector space with as a basis, the left multiplication extends to a representation, namely the regular representation of .
If is a representation, a subrepresentation is a subspace stable under the action of . A representation is irreducible if it does not contain a nontrivial subrepresentation (ie. different from itself and ).
If are two representations, we define the direct sum as the representation defined by the action for all , and .
Two representations and are isomorphic if there exists a -invariant isomorphism .
Our first result is:
Property 1: (Maschke) Any representation is a direct sum of irreducible representations.
Proof. Let be a finite group and be a representation. If is irreducible, there is nothing to prove, so we suppose that contains a nontrivial representation . Then, if is any scalar product on , it is possible to define a scalar product invariant under , for example by
Therefore, because is stable under , the orthogonal for so is, and . Finally, Property 1 follows by induction on the dimension of .
Of course, in the decomposition given by Property 1, the same representation may appear several times (up to isomorphism). Thanks to a theorem due to Frobenius, this number may be characterized.
Definitions: Let be a finite group. Let denote the set of central functions , that is the functions satisfying for all . We endow with the scalar product
Clearly, the dimension of is , the number of conjugacy classes of (a basis of is given by the set of functions taking the value on a fixed conjugacy class and otherwise).
Associated to a representation , we define a character defined by . A character is said irreducible if the associated representation so is.
Property 2: (Frobenius) Let be a finite group. The set of irreducible characters defines an orthonormal basis of .
Proof. We first prove that the family of irreducible characters is orthonormal. Of course, two isomorphic representations are conjugated, so their characters are equal; therefore, it is sufficient to prove that two irreducible representations satisfy if and are isomorphic and otherwise.
We define the representation as the representation defined by the action for every , and . Now, let denote the set of fixed points of the action of on .
Claim 1: (Schur) if and are isomorphic, and otherwise.
A nontrivial element of defines a -invariant isomorphism , hence if and are not isomorphic. From now on, suppose that and are isomorphic, and let be a -invariant isomorphism.
Let . Then . Let be an eigenvalue of , so that is not invertible. Notice that is a subrepresentation of , so because is irreducible and is not invertible. We deduce that that is .
We just proved that is one-dimensional.
Therefore, it is sufficient to prove that . We left as an exercise that . In particular, we deduce that . Now,
Claim 2: For any representation , where denotes the set of fixed points.
Let . It is easy to show that for all , so that we deduce that . Therefore, . Now we notice that . Indeed, if then for all , hence ; conversely, if , hence . We conclude that
Finally, from Claim 1 and Claim 2, we deduce that the family of irreducible characters is orthonormal. Now, if denotes the subspace of spanning by the irreducible characters, we want to prove that or equivalently that .
So let . We want to prove that, for any representation , the function is zero. According to Property 1, is a direct sum of irreducible representations ; in particular, . Therefore, we may suppose without loss of generality that is irreducible.
Now, it is easy to show that for every , so that . But and according to Claim 1, so for some . Now,
hence . Now, we apply this result for the regular representation : for every ,
hence . We just prove that , so .
We now are able to precise Property 1:
Property 3: Let be a finite group and let denote its irreducible representations up to isomorphism. If is a representations of , then
Proof. According to Property 1, may be written as a direct sum of , say . Then because . But we saw during the proof of Property 2 that if and are isomorphic and otherwise. Therefore, is the number of representations in the decomposition isomorphic to . So Property 3 follows.
Property 3 is our main result from representation theory, and now we are able to prove the two following corollaries:
Corollary 4: Let be a finite group. The number of conjugacy classes is equal to the number of irreducible representations up to isomorphism.
Proof. We already noticed that isomorphic representations lead to equal characters. Conversely, Property 3 implies that two representations with the same character are isomorphic. Therefore, is equal to the dimension of , is equal to the number of irreducible characters according to Property 2, and now the number of irreducible characters is equal to the number of irreducible representations.
Corollary 5: Let be a finite group. Then where the sum is taken over the set of irreducible representations up to isomorphism.
Proof. Let be the regular representation. Let be the decomposition given by Property 3. Then
Of course, so it is sufficient to prove that . Notice that is a permutation matrix, so that is the number of fixed points of the function from to itself; therefore, if and otherwise. Hence
Corollary 5 follows.
Now we are ready to prove Theorem 3:
Proof of Theorem 3. Let denote the irreducible representations of up to isomorphism. Representations of degree 1 are only morphisms from to , so the number of such representations is equal to the number of morphisms from the abelian group to .
But this number turns out to be because the number of morphisms from a finite abelian group to is precisely ; we leave this statement as an exercise (hint: argue by induction on the number of factors in the decomposition of as a direct sum of cyclic groups).
Therefore, the formula given by Corollary 5 may be written as
where . Therefore,
and Theorem 3 follows.
In fact, the relation
proved above may be quite useful to compute for some small groups. For example, if denotes the quaternion group, its commutator subgroup is , so the relation above becomes ; therefore, according to Corollary 4, has five conjugacy classes, hence . (Noticing that is not abelian, we deduce that Theorem 1 is optimal.)
Another example is the symmetric group : its commutator subgroup is , so the relation above becomes ; therefore, has three conjugacy classes, hence . (Noticing that is not nilpotent, since , we deduce that Theorem 2 is optimal; however, a lot a nilpotent groups satisfies because we saw that a group satisfying has nilpotent-length at most two.)
For more information on commutativity degree of finite groups, see Anna Castelaz’s thesis Commutativity degree of finite groups. In particular, an elementary proof of Theorem 1 based on conjugacy class equation can be found there, as well as a complete classification of groups satisfying (see Proposition 5.1.4):
Theorem: Let be a finite group. If , then either
- is abelian,
- or for some abelian group and some -group (in this case, ),
- or where is an abelian group, and