A SURVEY ON THE ESTIMATION OF COMMUTATIVITY IN FINITE GROUPS

A. K. DAS, R. K. NATH, AND M. R. POURNAKI

Abstract. Let G be a finite group and let C = {(x, y) ∈ G × G | xy = yx}. Then Pr(G) = |C|/|G|^{2} is the probability that two elements of G, chosen randomly with replacement, commute. This probability is a well known quantity, called commutativity degree of G, and indeed gives us an estimation of commutativity in G. In the last four decades this subject has enjoyed a flourishing development. In this article, we give a brief survey on the development of this subject and then we collect several of our results concerning Pr(G) as well as its various generalizations.

1. Introduction

The study of group theoretical problems related to discrete probability has long been a topic of interest. Indeed, the story goes back to a sequence of papers all of them entitled “On some problems of a statistical group-theory” written by P. Erdős and P. Turán during the years 1965 to 1972 (see [7, 8, 9, 10, 11, 12, 13]). In Theorem IV of [10] Erdős and Turán have shown that in an arbitrary finite group G, the number of elements of

C = {(x, y) ∈ G × G | xy = yx}

is equal to |G|k(G), where k(G) is the number of conjugacy classes of G. Later, in 1973, W. H. Gustafson [18] reproved this theorem adopting the same technique which was used by Erdős and Turán [10]. Here, for convenience of the readers, we reproduce the proof as follows:

Note that

C = ⨆_{x∈G} ({x} × C_{G}(x)),

where the symbol ⨆ denotes the disjoint union of sets and C_{G}(x) the centralizer of x in G. Therefore, we have

|C| = ∑_{x∈G} |{x} × C_{G}(x)| = ∑_{x∈G} |C_{G}(x)|.

Recall that if x and y are conjugate elements of G, then C_{G}(x) and C_{G}(y) are conjugate subgroups of G. Also the number of elements in the conjugacy class of x is equal to the index of C_{G}(x) in G, i.e., |G : C_{G}(x)|. Therefore, if x_{1}, …, x_{k(G)} are representatives of the distinct conjugacy classes in G, then we have

2000 Mathematics Subject Classification. Primary: 20A05, 20D60; Secondary: 20C15, 20P05.

Key words and phrases. Finite group, Group character, Commutativity degree.

The research of M. R. Pournaki was in part supported by a grant from the Academy of Sciences for the Developing World (TWAS–UNESCO Associateship – Ref. FR3240126591).

|C| = ∑_{i=1}^{k(G)} |G : C_{G}(x_{i})||C_{G}(x_{i})| = ∑_{i=1}^{k(G)} |G| = |G|k(G),

as shown by Erdős, Turán, and Gustafson.

Gustafson in [18] has also asked the following question which not only formed the title of his paper but also became the leading problem for the subsequent development of the subject:

What is the probability that two group elements commute?

Using the above-mentioned argument, Gustafson himself has provided an elementary but significant answer to this question. More precisely, if Pr(G) denotes the probability that two elements of G, chosen randomly with replacement, commute, then

Pr(G) = \frac{|C|}{|G|^{2}} = \frac{|G|k(G)}{|G|^{2}} = \frac{k(G)}{|G|}.

The quantity Pr(G) has been termed by some authors as the commutativity degree of G. Indeed, it gives us an estimation of commutativity in finite groups. Clearly, a finite group G is abelian if and only if Pr(G) = 1. Now, assuming G to be nonabelian, one can obtain (see [18]) an upper bound for Pr(G) as follows:

Consider the class equation of G given by

|G| = |Z(G)| + ∑_{i=1}^{t} |G : C_{G}(x_{i})|,

where Z(G) denotes the center of G and x_{1}, …, x_{t} are representatives of the distinct nontrivial conjugacy classes in G. For every i with 1 ≤ i ≤ t, we have |G : C_{G}(x_{i})| ≥ 2 and so t ≤ (|G| − |Z(G)|)/2. This implies that k(G) = |Z(G)| + t ≤ (|G| + |Z(G)|)/2. Since G is nonabelian, it is easy to see that G/Z(G) is not cyclic and so |G/Z(G)| ≥ 4. Therefore, it follows that

Pr(G) = \frac{k(G)}{|G|} ≤ \frac{|G| + |Z(G)|}{2|G|} = \frac{1}{2} + \frac{|Z(G)|}{2|G|} ≤ \frac{1}{2} + \frac{1}{8} = \frac{5}{8}.

Thus, in summary, given a finite group G, the probability Pr(G) that two of its elements, chosen randomly with replacement, commute equals k(G)/|G|, with Pr(G) = 1 if and only if G is abelian and Pr(G) ≤ 5/8 if G is nonabelian. Further, as the group of quaternions of order eight shows, the upper bound 5/8 is best possible. In [18], Gustafson has also shown, with a suitable interpretation, that this inequality remains valid for compact topological nonabelian groups. In the last four decades this subject has enjoyed a flourishing development. In the sequel we give a brief survey on the development of the subject.

In 1975, G. Sherman [33] generalized the above probability. Let G be a finite group acting on a finite set Ω. Put Pr_{G}(Ω) = |Fix(G, Ω)|/|G||Ω|, where Fix(G, Ω) is the set of pairs (g, ω) of G×Ω such that g.ω = ω. Note that Pr_{G}(Ω) is the probability that an element of G leaves an element of Ω fixed. Sherman [33] also considered the case for which G is abelian and A is the group of its automorphisms. Pr_{A}(G) is the probability that an automorphism leaves an element fixed. He proved that for groups of order p^{n}, p a prime, Pr_{A}(G) ≤ 2(3/p^{2})^{n/2}. He further showed that if G_{n} is a sequence of abelian groups with |G_{n}| → ∞, then Pr_{A}(G_{n}) → 0. Considering the action of G on itself by conjugation one obtains that Pr_{G}(G) is equal to Pr(G), i.e., the probability that two elements of G, chosen randomly with replacement, commute. In this sense Pr_{G}(Ω) is a generalization of Pr(G).

In 1979, D. J. Rusin [32] continued the study of Pr(G). He considered, for a finite group G, two situations, namely, G′ ⊆ Z(G) and G′ ∩ Z(G) = {1}, where G′ is the commutator subgroup of G. First, he obtained an explicit computation of Pr(G) for groups G with G′ ⊆ Z(G). In this case, G being the direct product of its Sylow subgroups, the product formula Pr(H × K) = Pr(H) Pr(K) sufficed him to obtain Theorem 1 of [32]: If G is a p-group with G′ ⊆ Z(G), then

Pr(G) = \frac{1}{|G′|}(1 + ∑_{G′/K cyclic} \frac{(p − 1)|G′ : K|}{p^{n(K)+1}}),

with n(K) defined by |{x ∈ G | gxg^{−1}x^{−1} ∈ K ∀g ∈ G}| = |G|/p^{n(K)}. In the second case, i.e., when G′ ∩ Z(G) = {1}, he showed that Pr(G) = Pr(K) holds true for some groups K such that K′ ≅ G′ and Z(K) = {1}. On the other hand, for every group G, there exist at most a finite number of groups K with K′ ≅ G′ and Z(K) = {1} (see [32, Proposition 4]). Under an additional condition that |G′| is a prime, he obtained an explicit computation of Pr(G) in this second case as well. In Section IV, Rusin has classified the groups G for which Pr(G) is greater than 11/32.

In 1995, P. Lescot [22] generalized Pr(G) in the other direction. Let G be a finite group and let n ≥ 0 be an integer. Define

d_{n}(G) = \frac{|\{(x_{1}, …, x_{n+1}) ∈ G^{n+1} | x_{i}x_{j} = x_{j}x_{i}, 1 ≤ i, j ≤ n + 1\}|}{|G|^{n+1}}.

Note that d_{1}(G) is equal to Pr(G), i.e., the probability that two elements of G, chosen randomly with replacement, commute. Therefore, this later quantity is a generalization of Pr(G). He showed that d_{n}(G) = d_{n}(H) provided G and H are isoclinic in the sense of P. Hall (see [19, Page 133]). He also showed that if G is a finite nonabelian group, then d_{n}(G) ≤ (3 × 2^{n} − 1)/2^{2n+1}. Moreover, the equality holds if and only if G is isoclinic to the group of quaternions of order eight. Also in 2001, Lescot [23] characterized all finite groups G with commutativity degree at least 1/2. This is done by first determining all finite groups G whose central factor is a nonabelian group of order pq and then using a result of S. R. Blackburn [2] determining all finite 2-groups with derived subgroup of order two.

In 2008, M. R. Pournaki and R. Sobhani [31] gave a new generalization of Pr(G). For a finite group G and g ∈ G define

Pr_{g}(G) = \frac{|\{(x, y) ∈ G × G | [x, y] = g\}|}{|G|^{2}},

where [x, y] denotes the commutator of x and y. Note that Pr_{1}(G) is equal to Pr(G), i.e., the probability that two elements of G, chosen randomly with replacement, commute. Therefore, this later quantity is a generalization of Pr(G). They found some results concerning Pr_{g}(G) and after that A. K. Das and R. K. Nath in a sequence of different papers gave new results and other new generalizations regarding Pr(G) and Pr_{g}(G) (see [3, 4, 5, 6, 28, 29]).

In the rest of this article we collect several of our results concerning Pr(G) as well as its various generalizations.

2. Commutativity of two elements

In this section we deal with probability that two elements of a finite group G, chosen randomly with replacement, commute. As we mentioned in Section 1, this probability is Pr(G) = |C|/|G|^{2}, where C = {(x, y) ∈ G × G | xy = yx}. Here we collect several results dealing with this probability.

In 1979, Rusin [32] computed, for a finite group G, the values of Pr(G) when G′ ⊆ Z(G), and also when G′ ∩Z(G) is trivial. In this direction we have the following result.

Theorem 2.1 ([5], Theorem 3.5). Let G be a finite group and let p be a prime number such that gcd(p − 1, |G|) = 1. If |G′| = p^{2} and |G′ ∩ Z(G)| = p, then the following statements hold:

(1) Pr(G) = \begin{cases}\frac{2p^{2} − 1}{p^{4}} & if C_{G}(G′) is abelian, \\ \frac{1}{p^{4}} (\frac{p − 1}{p^{2s−1}} + p^{2} + p − 1) & otherwise, \end{cases}

(2) |\frac{G}{Z(G)}| = \begin{cases}p^{3} & if C_{G}(G′) is abelian, \\ p^{2s+2} or p^{2s+3} & otherwise, \end{cases}

where p^{2s} = |C_{G}(G′) : Z(C_{G}(G′))|. Moreover, we have

|\frac{G}{G′ ∩ Z(G)} : Z(\frac{G}{G′ ∩ Z(G)})| = |\frac{G}{Z(G)} : Z(\frac{G}{Z(G)})| = p^{2}.

It may be noted here that, on a number of occasions, the structure of G/Z(G) is determined by its size. For example, using GAP [35] and the notion of semidirect product ‘⋊’, one can see that if G is a finite group with G′ ⊈ Z(G), then G/Z(G) is isomorphic to C_{7} ⋊ C_{3}, (C_{3} × C_{3}) ⋊ C_{3}, C_{13} ⋊ C_{3}, C_{19} ⋊ C_{3}, C_{3} × (C_{7} ⋊ C_{3}), or (C_{5} × C_{5}) ⋊ C_{3} according as |G/Z(G)| is equal to 21, 27, 39, 57, 63, or 75. Here, C_{n} denotes the cyclic group of order n.

Rusin also determined all numbers lying in the interval (11/32, 1] that can be realized as the commutativity degree of some finite groups, and also classified all finite groups whose commutativity degrees lie in the interval (11/32, 1]. In 2006, F. Barry, D. MacHale, and Á. Ní Shé [1] have shown that if G is a finite group with |G| odd and Pr(G) > 11/75, then G is supersolvable. They also proved that if Pr(G) > 1/3, then G is supersolvable. It may be mentioned here that a group G is said to be supersolvable if there is a series of the form

{1} = A_{0} ⊆ A_{1} ⊆ A_{2} ⊆ ⋯ ⊆ A_{r} = G,

where A_{i} ⊴ G and A_{i+1}/A_{i} is cyclic for every i with 0 ≤ i ≤ r − 1 (see [16, 24, 34]). The previous theorem enables us to obtain the following characterization.

Theorem 2.2 ([5], Theorem 4.3). Let G be a finite group. If |G| is odd and Pr(G) ≥ 11/75, then the possible values of Pr(G) and the corresponding structures of G′, G′ ∩ Z(G), and G/Z(G) are given by the following Table 1.

Pr(G)  G′  G′ ∩ Z(G)  G/Z(G)
1  {1}  {1}  {1}
\frac{1}{3} (1 + \frac{2}{3^{2s}})  C_{3}  C_{3}  (C_{3} × C_{3})^{s}, s ≥ 1
\frac{1}{5} (1 + \frac{4}{5^{2s}})  C_{5}  C_{5}  (C_{5} × C_{5})^{s}, s ≥ 1
\frac{5}{21}  C_{7}  \{1\}  C_{7} ⋊ C_{3}
\frac{55}{343}  C_{7}  C_{7}  C_{7} × C_{7}
\frac{17}{81}  C_{9} or C_{3} × C_{3}  C_{3} or  (C_{3} × C_{3}) ⋊ C_{3}
or C_{3} × C_{3}  C_{3} × C_{3}  or C_{3}^{3}
\frac{121}{729}  C_{3} × C_{3}  C_{3} × C_{3}  C_{3}^{4}
\frac{7}{39}  C_{13}  \{1\}  C_{13} ⋊ C_{3}
\frac{3}{19}  C_{19}  \{1\}  C_{19} ⋊ C_{3}
\frac{29}{189}  C_{21}  C_{3}  C_{3} × (C_{7} ⋊ C_{3})
\frac{11}{75}  C_{5} × C_{5}  \{1\}  (C_{5} × C_{5}) ⋊ C_{3}

Table 1. The possible values of Pr(G) and the corresponding structures of G′, G′ ∩ Z(G), and G/Z(G) provided |G| is odd and Pr(G) ≥ 11/75.

The following result gives a universal lower bound for Pr(G).

Theorem 2.3 ([28], Theorem 1). Let G be a finite group. Then we have

Pr(G) ≥ \frac{1}{|G′|} (1 + \frac{|G′| − 1}{|G : Z(G)|}) .

In particular, Pr(G) > 1/|G′| provided G is nonabelian.

There are several equivalent necessary as well as sufficient conditions for equality to hold in the above theorem. These are listed as follows.

Theorem 2.4 ([28], Theorem 2). Let G be a finite nonabelian group. Then the following statements are equivalent:

(1) The equality Pr(G) = \frac{1}{|G′|} (1 + \frac{|G′| − 1}{|G : Z(G)|}) holds.

(2) We have cd(G) = {1, |G : Z(G)|^{1/2}}, which means that G is of central type with |cd(G)| = 2. Here, cd(G) denotes the set of irreducible complex character degrees of G.

(3) For every x ∈ G \ Z(G), we have |Cℓ_{G}(x)| = |G′|. Here, Cℓ_{G}(x) denotes the conjugacy class of x.

(4) For every x ∈ G\Z(G), we have Cℓ_{G}(x) = G′x. In particular, G is a nilpotent group of class 2.

(5) For every x ∈ G\Z(G), we have C_{G}(x)⊴G and G′ ≅ G/C_{G}(x). In particular, G is a CN-group, i.e., a group in which the centralizer of every element is normal.

(6) For every x ∈ G \ Z(G), we have G′ = {[y, x] | y ∈ G}. In particular, every element of G′ is a commutator.

Theorem 2.3 and Theorem 2.4 not only allow us to obtain some characterizations for finite nilpotent groups of class 2 whose commutator subgroups have prime order, but also enable us to re-establish certain facts (essentially due to K. S. Joseph [21]) concerning the smallest prime divisors of the orders of finite groups.

Theorem 2.5 ([28], Proposition 1). Let G be a finite group and let p be the smallest prime divisor of |G|. Then the following statements hold:

(1) If p ≠ 2, then Pr(G) ≠ 1/p.

(2) In the case of G is nonabelian, Pr(G) > 1/p if and only if |G′| = p and G′ ⊆ Z(G).

Some of the consequences of the above results are as given below.

Theorem 2.6 ([28], Corollary 1). Let G be a finite group with Pr(G) = 1/3. Then |G| is even.

Theorem 2.7 ([28], Corollary 2). Let G be a finite group and let p ≠ 2 be the smallest prime divisor of |G|. If G is nonabelian with G′ ∩ Z(G) = {1}, then Pr(G) < 1/p.

For a group K and an element k of that group consider \overline{K} = K/Z(K) and \overline{k} = kZ(K), and define the map a_{K} : \overline{K} ×\overline{K} ⟶ K′ by a_{K}(\overline{x}, \overline{y}) = [x, y] which is obviously well defined. We now introduce the notion of isoclinism which is originally due to P. Hall (see [19, Page 133]). Let G and H be two groups. A pair (φ, ψ) is called an isoclinism from G to H if φ is an isomorphism from \overline{G} to \overline{H} and ψ is an isomorphism from G′ to H′ for which the following diagram commutes.

\overline{G} × \overline{G} ⟶^{φ×φ}  \overline{H} × \overline{H}
↓a_{G}  ↓a_{H}
G′  ⟶^{ψ}  H′

If there is an isoclinism from G to H, we shall say that G and H are isoclinic. Clearly, isoclinism is an equivalence relation between groups. It is well known that if G and H are isomorphic groups, then they are isoclinic.

Theorem 2.8 ([28], Proposition 2). Let G be a finite group and let p be a prime. Then the following statements are equivalent:

(1) We have |G′| = p and G′ ⊆ Z(G).

(2) The group G is of central type with |cd(G)| = 2 and |G′| = p.

(3) The group G is a direct product of a p-group P and an abelian group A such that |P′| = p and gcd(p, |A|) = 1.

(4) The group G is isoclinic to an extra-special p-group, and consequently, the equality |G : Z(G)| = p^{2k} holds true for some positive integer k.

In particular, if G is nonabelian and p is the smallest prime divisor of |G|, then the above statements are also equivalent to the condition Pr(G) > 1/p.

If |G′| = p, p prime, and G′ ⊆ Z(G), then we call the integer 2k + 1, where k is as in Part 4 of Theorem 2.8, as the isoclinic exponent of G and denote it by iso.exp(G). Thus, we have, in particular, the following theorem:

Theorem 2.9 ([31], Proposition 3.4). Let G be a finite group such that G′ ⊆ Z(G) and let |G′| = p be a prime. If iso.exp(G) = n, then

Pr(G) = \frac{1}{p} (1 + \frac{p − 1}{p^{n−1}}) .

Now consider a finite group G such that |G′| = p is a prime and G′ ∩ Z(G) = {1}. We have (G/Z(G))′ ≅ G′. On the other hand, if we consider Z(G/Z(G)) = H/Z(G) for some H, Z(G) ≤ H ≤ G, then [G, H] ≤ G′ ∩ Z(G) = {1} implies that H = Z(G) and so we have Z(G/Z(G)) = 1. Therefore by [32, Proposition 5], there is a positive integer n depending only on G for which G/Z(G) = ⟨a, b | a^{p} = b^{n} = 1, bab^{−1} = a^{r}⟩. We call n as the invariant number of G and denote it by inv(G).

Theorem 2.10 ([31], Proposition 4.4). Let G be a finite group such that G′ ∩Z(G) = {1} and let |G′| = p be a prime. If inv(G) = n, then

Pr(G) = \frac{n^{2} + p − 1}{pn^{2}}.

In [15, Corollary 1.2], I. V. Erovenko and B. Sury have shown, in particular, that for every integer k > 1 there exists a family {G_{n}} of finite groups such that Pr(G_{n}) → 1/k^{2} as n → ∞. In this connection, we have the following observations.

Theorem 2.11 ([27], Proposition 2.5.1). For any k ∈ ℕ with k > 1, there exists a family {G_{n}} of finite groups such that Pr(G_{n}) → 1/k as n → ∞.

Theorem 2.12 ([27], Proposition 2.5.2). For any n ∈ ℕ, there exists a finite group G such that Pr(G) = 1/n.

The following result gives us a formula as well as an upper bound for Pr(G) when G has exactly two irreducible complex character degrees.

Theorem 2.13 ([31], Theorem 2.2). Let G be a finite group such that cd(G) = {1, m}, m > 1. Then we have

Pr(G) = \frac{1}{|G′|} (1 + \frac{|G′| − 1}{m^{2}}) ≤ \frac{|G′| + 3}{4|G′|}.

3. Commutators of two elements which are equal to a given element

In this section we deal with the probability that the commutator of two elements, chosen randomly with replacement, in a finite group is equal to a given element of that group. For a finite group G and a given element g of G, this probability is Pr_{g}(G) = |C_{g}|/|G|^{2}, where C_{g} = {(x, y) ∈ G × G | [x, y] = g}. Clearly, this notion is a generalization of the pervious one, mentioned in Section 2, as we have Pr_{1}(G) = Pr(G).

In the following we collect several results dealing with this probability. Obviously for those g’s lie in G \ G′, we have Pr_{g}(G) = 0. Therefore in the sequel we deal only with g’s which lie in G′. Note that there are examples of groups G of order 96 due to R. M. Guralnick [17] where Pr_{g}(G) = 0 even when g belongs to G′.

The first result shows that Pr_{g}(G) is an invariant under isoclinism in the following sense.

Theorem 3.1 ([31], Lemma 3.5). Let G and H be two isoclinic finite groups and let (φ, ψ) be an isoclinism from G to H. If g ∈ G′, then we have

Pr_{g}(G) = Pr_{ψ(g)}(H).

The following result gives us a character theoretical formula for Pr_{g}(G).

ESTIMATION OF COMMUTATIVITY IN FINITE GROUPS 9

Theorem 3.2 ([31], Theorem 2.1). Let G be a finite group and g ∈ G′. Then

Pr_{g}(G) = \frac{1}{|G|} ∑_{χ∈Irr(G)} \frac{χ(g)}{χ(1)},

where Irr(G) denotes the set of irreducible complex characters of G.

We also derive a formula as well as a lower bound for Pr_{g}(G) when G has exactly two irreducible complex character degrees. Theorem 3.3 ([31], Theorem 2.2). Let G be a finite group with cd(G) = {1, m}, m > 1. If g ∈ G′, g ≠ 1, then we have

Pr_{g}(G) = \frac{1}{|G′|} (1 − \frac{1}{m^{2}} ) ≥ \frac{3}{4|G′|}.

In view of [20, Problem 2.13], we obtain the following theorem. Theorem 3.4 ([31], Propositions 3.1 and 3.4). Let G be a finite group such that G′ ⊆ Z(G) and let |G′| = p be a prime. If g ∈ G′, g ≠ 1, then we have

Pr_{g}(G) = \frac{1}{p} (1 − \frac{1}{|G : Z(G)|}) ≥ \frac{3}{4p}.

In particular, if iso.exp(G) = n, then we obtain that

Pr_{g}(G) = \frac{1}{p} (1 − \frac{1}{p^{n−1}} ) .

A universal upper bound for Pr_{g}(G), g ∈ G′, g ≠ 1, is given as follows. Theorem 3.5 ([31], Corollary 2.3). Let G be a finite group with |cd(G)| = 2. Let g ∈ G′, g ≠ 1. Then we have

Pr_{g}(G) ≤ \frac{1}{|G′|} (1 − \frac{1}{|G : Z(G)|}) .

Moreover, the equality holds if and only if G is of central type.

The following result gives us a formula for Pr_{g}(G), g ∈ G′, g ≠ 1, when |G′| is prime and G′ ∩ Z(G) is trivial. Theorem 3.6 ([31], Proposition 4.4). Let G be a finite group such that G′ ∩ Z(G) = {1} and let |G′| = p be a prime. If inv(G) = n and g ∈ G′, g ≠ 1, then we have

Pr_{g}(G) = \frac{n^{2} − 1}{pn^{2}}.

We close this section by listing some more properties of Pr_{g}(G). Theorem 3.7 ([31], Proposition 5.1). Let G be a finite group and let g ∈ G′. Then we have Pr_{g}(G) ≤ Pr(G). Moreover, the equality holds if and only if g = 1.

Theorem 3.8 ([31], Proposition 5.2). Let G be a finite group and let g ∈ G′, g ≠ 1. Then we have Pr_{g}(G) < 1/2.

Theorem 3.9 ([31], Proposition 5.3). For any ϵ ∈ ℝ with ϵ > 0, there exists a finite group G and g ∈ G such that 1/2 − ϵ < Pr_{g}(G) < 1/2.

4. Admissible words which are equal to a given element

In this section we deal with a probability which is a generalization of the previous one mentioned in Section 3. For a given positive integer n, consider the free group of words on n generators x_{1}, x_{2}, …, x_{n}. A word ω(x_{1}, x_{2}, …, x_{n}) is called admissible (see [3]) if each letter in it has precisely two nonzero indices, namely, +1 and −1. Given a nontrivial admissible word ω(x_{1}, x_{2}, …, x_{n}), n ≥ 2, and an element g ∈ G, consider the ratio

Pr_{g}^{ω}(G) = \frac{|\{(g_{1}, g_{2}, …, g_{n}) ∈ G^{n} | ω(g_{1}, g_{2}, …, g_{n}) = g\}|}{|G^{n}|}.

Note that, for ω(x_{1}, x_{2}) = x_{1}x_{2}x_{1}^{−1}x_{2}^{−1}, we have Pr_{g}^{ω}(G) = Pr_{g}(G), and so this later probability is a generalization of Pr_{g}(G).

In the following we collect several results dealing with this probability.

Theorem 4.1 ([6], Proposition 2.2). Let G be a finite group and let ω(x_{1}, x_{2}, …, x_{n}) be a nontrivial admissible word. Then the following statements hold:

(1) Pr_{1}^{ω}(G) = 1 if and only if G is abelian.

(2) Pr_{g}^{ω}(G) = Pr_{h}^{ω}(G) if g, h ∈ G′ are conjugate in G.

(3) Pr_{g}^{ω}(G) = Pr_{h}^{ω}(G) if g, h ∈ G′ generate the same cyclic subgroups of G. Consequently, we obtain that

Pr_{g}^{ω}(G) = \frac{1 − Pr_{1}^{ω}(G)}{p − 1}

provided |G′| = p is a prime and g ∈ G′, g ≠ 1.

The quantity Pr_{g}^{ω}(G) respects the cartesian product in the following sense.

Theorem 4.2 ([6], Proposition 2.3). Let H and K be two finite groups, let (h, k) ∈ H′ × K′ and ω(x_{1}, x_{2}, …, x_{n}) be a nontrivial admissible word. Then we have

Pr_{(h,k)}^{ω}(H × K) = Pr_{h}^{ω}(H)Pr_{k}^{ω}(K).

The following result shows that Pr_{g}^{ω}(G) is an invariant under isoclinism of finite groups.

Theorem 4.3 ([6], Proposition 2.4). Let G and H be two finite groups and (ϕ, ψ) be an isoclinism from G to H. If g ∈ G′ and ω(x_{1}, x_{2}, …, x_{n}) is a nontrivial admissible word, then we have

Pr_{g}^{ω}(G) = Pr_{ψ(g)}^{ω}(H).

Theorem 4.4 ([6], Proposition 3.1). Let G be a finite group and let ω(x_{1}, x_{2}, …, x_{n}) be a nontrivial admissible word. If g ∈ G′, then we have

\frac{Pr_{g}(G)}{|G : Z(G)|^{n−2}} ≤ Pr_{g}^{ω}(G).

If G is a finite nonabelian simple group and ω(x_{1}, x_{2}, …, x_{n}) is a nontrivial admissible word, then using Ore conjecture (see [30]), which has been established recently in [25, Theorem 1], it follows from the above result that every element of G is of the form ω(g_{1}, g_{2}, …, g_{n}) for some g_{1}, g_{2}, …, g_{n} ∈ G.

As a generalization of Theorem 3.7, we have the following result.

Theorem 4.5 ([6], Proposition 3.6). Let G be a finite group and let ω(x_{1}, x_{2}, …, x_{n}) be a nontrivial admissible word. If g ∈ G′, then the following statements hold:

(1) Pr_{g}^{ω}(G) ≤ Pr_{1}^{ω}(G) ≤ Pr(G).

(2) Pr_{g}^{ω}(G) = Pr_{1}^{ω}(G) if and only if g = 1.

Consequently, Pr_{g}^{ω}(G) = 1 if and only if g = 1 and G is abelian.

The following result gives us a universal lower bound for Pr_{1}^{ω}(G).

Theorem 4.6 ([6], Proposition 3.7). Let G be a finite group and let ω(x_{1}, x_{2}, …, x_{n}) be a nontrivial admissible word. Then we have

Pr_{1}^{ω}(G) ≥ \frac{1}{|G′|} (1 + \frac{|G′| − 1}{|G : Z(G)|^{n/2}} ) .

In particular, Pr_{1}^{ω}(G) > \frac{1}{|G′|} provided G is nonabelian.

The following result generalizes Theorem 3.8.

Theorem 4.7 ([6], Proposition 3.8). Let G be a finite nonabelian group, let g ∈ G′ and p be the smallest prime divisor of |G|. If ω(x_{1}, x_{2}, …, x_{n}) is a nontrivial admissible word and g ≠ 1, then Pr_{g}^{ω}(G) < 1/p. In particular, we have Pr_{g}^{ω}(G) < 1/2.

Let us now take the nontrivial admissible word ω(x_{1}, x_{2}, …, x_{n}) to be x_{1}x_{2}…x_{n}x_{1}^{−1} x_{2}^{−1}…x_{n}^{−1}, and write Pr_{g}^{n}(G) in place of Pr_{g}^{ω}(G). It has been observed that Pr_{g}^{n}(G) = Pr_{g}^{n+1}(G). Hence, without any loss we may assume that n is even.

Theorem 4.8 ([29], Equations 8, 11, and 12). Let G be a finite nonabelian group, let g ∈ G′ and 2 ≤ d ≤ min{χ(1) | χ ∈ Irr(G), χ(1) ≠ 1}. Then the following statements hold:

(1) |Pr_{g}^{n}(G) − \frac{1}{|G′|}| ≤ \frac{1}{d^{n−2}} (Pr(G) − \frac{1}{|G′|}).

(2) |Pr_{g}^{n}(G) − \frac{1}{|G′|}| ≤ \frac{1}{d^{n}} (1 − \frac{1}{|G′|}). In particular, we have

Pr_{g}^{n}(G) ≤ \frac{2^{n} + 1}{2^{n+1}}.

Theorem 4.9 ([29], Equation 14). Let G be a finite nonabelian simple group and let g ∈ G. Then we have

|Pr_{g}^{n}(G) − \frac{1}{|G|}| ≤ \frac{1}{3^{n−2}} (\frac{1}{12} − \frac{1}{|G|}) .

In particular, we conclude that

Pr_{g}^{n}(G) ≤ \frac{3^{n−2} + 4}{3^{n−1} × 20}.

The following result generalizes Theorem 3.3.

Theorem 4.10 ([29], Proposition 5.1). Let G be a finite nonabelian group and let g ∈ G′, g ≠ 1. If cd(G) = {1, d}, d > 1, then we have

Pr_{g}^{n}(G) = \frac{1}{|G′|} (1 − \frac{1}{d^{n}} ) .

The next two results give us some necessary and sufficient conditions for equality to hold in Theorem 4.8.

Theorem 4.11 ([29], Propositions 4.2 and 5.2). Let G be a finite nonabelian group, let g ∈ G′ and 2 ≤ d ≤ min{χ(1) | χ ∈ Irr(G), χ(1) ≠ 1}. Then the following statements hold:

(1) Pr_{g}^{n}(G) = \frac{1}{d^{n−2}} (Pr(G) + \frac{d^{n−2} − 1}{|G′|}) if and only if

g = 1 and cd(G) = {1, d}.

(2) Pr_{g}^{n}(G) = \frac{1}{d^{n−2}} (− Pr(G) + \frac{d^{n−2} + 1}{|G′|}) if and only if

g ≠ 1, cd(G) = {1, d}, and |G′| = 2.

Theorem 4.12 ([29], Proposition 4.4 and Corollary 5.3). Let G be a finite nonabelian group, let g ∈ G′ and 2 ≤ d ≤ min{χ(1) | χ ∈ Irr(G), χ(1) ≠ 1}. Then the following statements hold:

(1) Pr_{g}^{n}(G) = \frac{1}{d^{n}} (1 + \frac{d^{n} − 1}{|G′|}) if and only if

g = 1 and cd(G) = {1, d}.

(2) Pr_{g}^{n}(G) = \frac{1}{d^{n}} (−1 + \frac{d^{n} + 1}{|G′|}) if and only if

g ≠ 1, cd(G) = {1, d}, and |G′| = 2.

The following result provides a characterization of finite groups up to isoclinism.

Theorem 4.13 ([29], Proposition 4.6). Let G be a finite nonabelian group, let g ∈ G′ and p be the smallest prime divisor of |G|. Then we have

Pr_{g}^{n}(G) = \frac{p^{n} + p − 1}{p^{n+1}},

if and only if g = 1 and G is isoclinic to

⟨x, y | x^{p^{2}} = 1 = y^{p}, y^{−1}xy = x^{p+1}⟩.

In particular, putting p = 2, we conclude that

Pr_{g}^{n}(G) = \frac{2^{n} + 1}{2^{n+1}},

if and only if g = 1 and G is isoclinic to D_{8}, the dihedral group, and hence, to Q_{8}, the group of quaternions.

We now list a few results which are basically generalizations of some of the results obtained in [31].

Theorem 4.14 ([29], Proposition 6.1). Let G be a finite nonabelian group with |cd(G)| = 2 and let g ∈ G′. Then we have

Pr_{1}^{n}(G) ≥ \frac{1}{|G′|} (1 + \frac{|G′| − 1}{|G : Z(G)|^{n/2}} ) and

Pr_{g}^{n}(G) ≤ \frac{1}{|G′|} (1 − \frac{1}{|G : Z(G)|^{n/2}} ) provided g ≠ 1.

Moreover, in each case, the equality holds if and only if G is of central type.

Theorem 4.15 ([29], Corollary 6.2). Let G be a finite nonabelian group and g ∈ G′. Let G be of central type with |cd(G)| = 2. Then we have

Pr_{1}^{n}(G) ≤ \frac{1}{|G′|} (1 + \frac{|G′| − 1}{2^{n}}) and

Pr_{g}^{n}(G) ≥ \frac{1}{|G′|} (1 − \frac{1}{2^{n}} ) provided g ≠ 1.

Theorem 4.16 ([29], Proposition 6.3). Let G be a finite nonabelian group and let G′ ⊆ Z(G) and |G′| = p be a prime. If g ∈ G′, then we have

Pr_{g}^{n}(G) = \begin{cases}\frac{1}{p} (1 + \frac{p − 1}{p^{nk}}) & if g = 1, \\ \frac{1}{p} (1 − \frac{1}{p^{nk}} ) & if g ≠ 1, \end{cases}

where k = \frac{1}{2} log_{p} |G : Z(G)|.

Theorem 4.17 ([29], Proposition 6.5). Let G be a finite nonabelian group and g ∈ G′. If G′ ∩ Z(G) = {1}, |G′| = p, where p is a prime, and inv(G) = r, then we have

Pr_{g}^{n}(G) = \begin{cases}\frac{r^{n} + p − 1}{pr^{n}} & if g = 1, \\ \frac{r^{n} − 1}{pr^{n}} & if g ≠ 1. \end{cases}

Theorem 4.18 ([29], Proposition 6.7). For any ε ∈ ℝ with ε > 0 and for any prime number p, there exists a finite group G such that the inequality

|Pr_{g}^{n}(G) − \frac{1}{p}| < ε

holds true for all g ∈ G′.

5. Commutators of two subgroups which are equal to a given element

In 2007, A. Erfanian, R. Rezaei, and P. Lescot [14] studied the probability Pr(H, G) that an element of a given subgroup H of a finite group G commutes with an element of G (see also [26]). Note that Pr(G, G) = Pr(G). This notion has been further generalized as follows. Let G be a finite group and g ∈ G′. Let H and K be two subgroups of G. Consider the ratio

Pr_{g}(H, K) = \frac{|\{(x, y) ∈ H × K | [x, y] = g\}|}{|H||K|}.

If g = 1, then for brevity we write Pr_{1}(H, K) = Pr(H, K). Note that for H = K = G, we have Pr_{g}(H, K) = Pr_{g}(G).

The following theorem says that Pr_{g}(H, K) is not very far from being symmetric with respect to H and K.

Theorem 5.1 ([4], Proposition 2.1). Let G be a finite group and let g ∈ G′. If H and K are two subgroups of G, then we have

Pr_{g}(H, K) = Pr_{g^{−1}}(K, H).

However, if g^{2} = 1, or, if g ∈ H ∪ K (for example, when H or K is normal in G), then we have

Pr_{g}(H, K) = Pr_{g}(K, H) = Pr_{g^{−1}}(H, K).

The quantity Pr_{g}(H, K) respects the cartesian product in the following sense.

Theorem 5.2 ([4], Proposition 2.2). Let G_{1} and G_{2} be two finite groups with subgroups H_{1}, K_{1} ⊆ G_{1} and H_{2}, K_{2} ⊆ G_{2}. Let g_{1} ∈ G′_{1} and g_{2} ∈ G′_{2}. Then we have

Pr_{(g_{1},g_{2})}(H_{1} × H_{2}, K_{1} × K_{2}) = Pr_{g_{1}}(H_{1}, K_{1})Pr_{g_{2}}(H_{2}, K_{2}).

Now, we have the following computing formula which plays a key role in the study of Pr_{g}(H, K).

Theorem 5.3 ([4], Theorem 2.3). Let G be a finite group and let g ∈ G′. If H and K are two subgroups of G, then we have

Pr_{g}(H, K) = \frac{1}{|H||K|}_{∑_{x∈H} ; g^{−1}x∈Cℓ_{K}(x)}|C_{K}(x)| = \frac{1}{|H|}_{∑_{x∈H} ; g^{−1}x∈Cℓ_{K}(x)} \frac{1}{|Cℓ_{K}(x)|},

where C_{K}(x) = {y ∈ K | xy = yx} and Cℓ_{K}(x) = {yxy^{−1} | y ∈ K}, the K−conjugacy class of x.

The following result generalizes the well known formula Pr(G) = k(G)/|G|.

Theorem 5.4 ([4], Corollary 2.4). Let G be a finite group and let H and K be two subgroups of G. If H ⊴ G, then we have

Pr(H, K) = \frac{k_{K}(H)}{|H|},

where k_{K}(H) is the number of K−conjugacy classes that constitute H.

Let G be a finite group. If H ⊴ G with C_{G}(x) ⊆ H for all x ∈ H \ {1}, then using Sylow’s theorems and the fact that nontrivial p-groups have nontrivial centers, we have gcd(|H|, |G : H|) = 1. Therefore, by the Schur-Zassenhaus theorem, H has a complement in G. Such groups belong to a well known class of groups called the Frobenius groups; for example, the alternating group A_{4}, the dihedral groups of order 2n with n odd, the nonabelian groups of order pq, where p and q are primes with q|(p − 1), etc.

Theorem 5.5 ([4], Proposition 2.5). Let G be a finite group. If H is an abelian normal subgroup of G with a complement K in G and g ∈ G′, then we have

Pr_{g}(H, G) = Pr_{g}(H, K).

As a consequence, we conclude the following theorem.

Theorem 5.6 ([4], Corollary 2.6). Let G be a finite group and let g ∈ G′. If H ⊴ G with C_{G}(x) = H for all x ∈ H \ {1}, then we have

Pr_{g}(H, G) = Pr_{g}(H, K),

where K is a complement of H in G. In particular,

Pr(H, G) = \frac{1}{|H|} + \frac{|H| − 1}{|G|}.

The following result gives us some conditional lower bounds for Pr_{g}(H, K).

Theorem 5.7 ([4], Proposition 3.1). Let G be a finite group and g ∈ G′. Let H and K be any two subgroups of G. If g ≠ 1, then the following statements hold:

(1) If Pr_{g}(H, K) ≠ 0 then Pr_{g}(H, K) ≥ \frac{|C_{H}(K)||C_{K}(H)|}{|H||K|}.

(2) If Pr_{g}(H, G) ≠ 0 then Pr_{g}(H, G) ≥ \frac{2|H ∩ Z(G)||C_{G}(H)|}{|H||G|}.

(3) If Pr_{g}(G) ≠ 0 then Pr_{g}(G) ≥ \frac{3}{|G : Z(G)|^{2}} .

The next two results are generalizations of Theorem 3.7 and Theorem 3.8.

Theorem 5.8 ([4], Proposition 3.2). Let G be a finite group and let g ∈ G′. If H and K are any two subgroups of G, then

Pr_{g}(H, K) ≤ Pr(H, K).

Moreover, the equality holds if and only if g = 1.

Theorem 5.9 ([4], Proposition 3.3). Let G be a finite group and g ∈ G′, g ≠ 1. Let H and K be any two subgroups of G. If p is the smallest prime divisor of |G|, then we have

Pr_{g}(H, K) ≤ \frac{|H| − |C_{H}(K)|}{p|H|} < \frac{1}{p}.

The quantity Pr_{g}(H, K) is monotonic in the following sense.

Theorem 5.10 ([4], Proposition 3.4). Let G be a finite group. Let H, K_{1}, and K_{2} be three subgroups of G with K_{1} ⊆ K_{2}. Then we have

Pr(H, K_{1}) ≥ Pr(H, K_{2}).

Moreover, the equality holds if and only if Cℓ_{K_{1}}(x) = Cℓ_{K_{2}}(x) for all x ∈ H.

Theorem 5.11 ([4], Proposition 3.5). Let G be a finite group. Let H, K_{1}, and K_{2} be three subgroups of G with K_{1} ⊆ K_{2}. Then we have

Pr(H, K_{2}) ≥ \frac{1}{|K_{2} : K_{1}|} (Pr(H, K_{1}) + \frac{|K_{2}| − |K_{1}|}{|H||K_{1}|}) .

Moreover, the equality holds if and only if C_{H}(x) = {1} for all x ∈ K_{2} \ K_{1}.

Theorem 5.12 ([4], Proposition 3.6). Let G be a finite group. Let H_{1} ⊆ H_{2} and K_{1} ⊆ K_{2} be subgroups of G and g ∈ G′. Then we have

Pr_{g}(H_{1}, K_{1}) ≤ |H_{2} : H_{1}||K_{2} : K_{1}|Pr_{g}(H_{2}, K_{2}).

Moreover, the equality holds if and only if

g^{−1}x ∉ Cℓ_{K_{2}}(x) for all x ∈ H_{2} \ H_{1},
g^{−1}x ∉ Cℓ_{K_{2}}(x) \ Cℓ_{K_{1}}(x) for all x ∈ H_{1},
and C_{K_{1}}(x) = C_{K_{2}}(x) for all x ∈ H_{1} with g^{−1}x ∈ Cℓ_{K_{1}}(x).

In particular, for g = 1, the condition for equality reduces to H_{1} = H_{2} and K_{1} = K_{2}.

The following result also generalizes Theorem 3.7 in some sense.

Theorem 5.13 ([4], Corollary 3.7). Let G be a finite group, let H be a subgroup of G and g ∈ G′. Then we have

Pr_{g}(H, G) ≤ |G : H| Pr(G).

Moreover, the equality holds if and only if g = 1 and H = G.

We continue the survey by mentioning a few generalizations of some results obtained in [14].

Theorem 5.14 ([4], Theorem 3.8). Let G be a finite group and let p be the smallest prime divisor of |G|. If H and K are any two subgroups of G, then we have

Pr(H, K) ≥ \frac{|C_{H}(K)|}{|H|} + \frac{p(|H| − |X_{H}| − |C_{H}(K)|) + |X_{H}|}{|H||K|} and

Pr(H, K) ≤ \frac{(p − 1)|C_{H}(K)| + |H|}{p|H|} − \frac{|X_{H}|(|K| − p)}{p|H||K|},

where X_{H} = {x ∈ H | C_{K}(x) = {1}}. Moreover, in each of these bounds, H and K can be interchanged.

Theorem 5.15 ([4], Corollary 3.9). Let G be a finite group and let p be the smallest prime divisor of |G|. If H and K are two subgroups of G such that [H, K] ≠ {1}, then we have

Pr(H, K) ≤ \frac{2p − 1}{p^{2}}.

In particular, we conclude that Pr(H, K) ≤ 3/4.

Theorem 5.16 ([4], Proposition 3.10). Let G be a finite group and let H and K be any two subgroups of G. If Pr(H, K) = (2p − 1)/p^{2} for some prime p, then p divides |G|. If p happens to be the smallest prime divisor of |G|, then we have

\frac{H}{C_{H}(K)} ≅ C_{p} ≅ \frac{K}{C_{K}(H)}, and hence, H ≠ K.

In particular, we conclude that

\frac{H}{C_{H}(K)} ≅ C_{2} ≅ \frac{K}{C_{K}(H)} provided Pr(H, K) = \frac{3}{4}.

We conclude the survey with the following result, which gives us a character theoretical formula.

Theorem 5.17 ([4], Theorem 4.2 and Proposition 4.4). Let G be a finite group. If H is a normal subgroup of G and g ∈ G′, then we have

Pr_{g}(H, G) = \frac{1}{|G|} ∑_{χ∈Irr(G)} \frac{[χ_{H}, χ_{H}]}{χ(1)}χ(g).

Consequently, we conclude that

|Pr_{g}(H, G) − \frac{1}{|G′|}| ≤ |G : H| (Pr(G) − \frac{1}{|G′|}) .

The above result yields, in particular, that if G is a finite group with |G′| ≤ p^{2}, where p is the smallest prime divisor of |G|, then every element of G′ is a commutator.

Acknowledgments

Part of this work was done while the third author visited the Delhi Center of the Indian Statistical Institute (ISID). He would like to thank the Academy of Sciences for the Developing World (TWAS) and ISID for sponsoring his visits to New Delhi in July–August 2007 and January 2010. Especially he would like to express his thanks to Professor Rajendra Bhatia for the hospitality enjoyed at ISID.

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A. K. Das, Department of Mathematics, North-Eastern Hill University, Permanent Campus, Shillong-793022, Meghalaya, India.

E-mail address: akdasnehu@gmail.com

R. K. Nath, Department of Mathematics, North-Eastern Hill University, Permanent Campus, Shillong-793022, Meghalaya, India.

E-mail address: rajatkantinath@yahoo.com

M. R. Pournaki, Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran.

E-mail address: pournaki@ipm.ir
URL: http://math.ipm.ac.ir/pournaki/