On Commutators in Groups

Commutators originated over 100 years ago as a by-product of computing group characters of nonabelian groups. They are now an established and immensely useful tool in all of group theory. Commutators became objects of interest in their own right soon after their introduction. In particular, the phenomenon that the set of commutators does not necessarily form a subgroup has been well documented with various kinds of examples. Many of the early results have been forgotten and were rediscovered over the years. In this paper we give a historical overview of the origins of commutators and a survey of different kinds of groups where the set of commutators does not equal the commutator subgroup. We conclude with a status report on what is now called the Ore Conjecture stating that every element in a finite nonabelian simple group is a commutator. 1 Origins of commutators “In a group the product of two commutators need not be a commutator, consequently the commutator group of a given group cannot be defined as the set of all commutators, but only as the group generated by these. There seems to exist very little in the way of criteria or investigations on the question when all elements of the commutator group are commutators.” This is what Oystein Ore says in 1951 in the introduction to his paper “Some remarks on commutators” [57]. Since Ore made his comments, numerous contributions have been made to this topic and they are widely scattered over the literature. Many results have been rediscovered and republished. A case in point is Ore himself. The main result of [57] is that the alternating group on n letters, n ≥ 5, consists entirely of commutators. This was already proved by G. A. Miller [54] over half a century earlier. The two authors of this paper almost got into a similar situation after rediscovering one of the major results in this area. Fortunately we realized this before publication and then concluded that a survey of the major questions and results in this area was needed, together with a historical overview of the origins of commutators. Kappe, Morse: On Commutators in groups 2 Commutators came into the world 125 years ago as a by-product of Dedekind’s first foray into determining group characters of nonabelian groups. In an 1896 letter to Frobenius, Dedekind revealed his ideas and results for the first time. Here is what Frobenius says in [16]: “Das Element F , das sich mittelst der Gleichung BA = ABF aus A und B ergiebt, nenne ich nach DEDEKIND den Commutator von A und B.” 1 According to Frobenius, Dedekind proved in 1880 that the conjugate of a commutator is again a commutator, and therefore that the commutator subgroup generated by the commutators of a group is a normal subgroup of the group. Furthermore, Dedekind proved that any normal subgroup with abelian quotient contains the commutator subgroup, and that the commutator subgroup is trivial if and only if the group is abelian. However, these results were first published by G. A. Miller in [52]. The motivating force behind Dedekind’s introduction of commutators was his goal of extending group characters from abelian to nonabelian groups. The central object of investigation for Frobenius and Dedekind was the group determinant and its factorization, out of which arose the theory of group characters. For the definition of the group determinant and further details we refer to [9], since we are only interested in a by-product of this concept, namely commutators. Dedekind had spent the early years of his career at the ETH Zürich (1858-62). In 1880 he revisited Zürich and became personally acquainted with Frobenius, 18 years his junior, who was at the time a professor at the ETH. This was the starting point of an on-again-off-again, sometimes intense, correspondence between the two over many years as detailed by Hawkins in [32] and [33]. Around 1880 Dedekind was motivated by his studies of the discriminant in a normal field to consider the group determinant. One of the earliest results he obtained was that for a finite abelian group of order n the group determinant factored into n linear factors with the characters as coefficients of the linear factors. In his correspondence with Frobenius, Dedekind conjectured that for a nonabelian group G the number of linear factors of the group determinant was equal to the index of the commutator subgroup G′ in G, with coefficients corresponding to those of the abelian group G/G′, and in this context commutators and the commutator subgroup made their appearance. A good deal of the correspondence between Dedekind and Frobenius deals with the group determinant, its factorization, and Dedekind’s conjecture stated above. Dedekind determined the group determinant and its factorization for the symmetric group S3 and the quaternions of order 8, and in turn, Frobenius did the same for the dihedral group of order 8. Finally, in his 1896 paper [16] Frobenius proves Dedekind’s conjecture as part of the general theorem on the factorization of the group determinant for finite nonabelian groups. For the details of this result we refer the interested reader to Theorem 3.4 in [9]. “Following Dedekind, I am calling the element F , which is obtained from A and B with the help of the equation BA = ABF , the commutator of A and B.” Kappe, Morse: On Commutators in groups 3 Dedekind himself never published anything concerning the group determinant nor its connection with commutators. However, according to Hawkins [33], Dedekind decided to pursue some group theoretic research of his own that allowed him to use his commutators. Earlier, Dedekind had studied normal extensions of the rational field with all subfields normal. Some years later these investigations suggested to him the related problem: Characterize those groups with the property that all subgroups are normal – he called such groups Hamiltonian. Dedekind found, by making use of commutators, that determining the answer was relatively simple, and he communicated this to his friend Heinrich Weber, an editor of the Mathematische Annalen, who urged him to publish the result there. Dedekind eventually published his results in [11], but only after checking with Frobenius, who assured him that this result was significant and not a consequence of known results. As was already mentioned, G. A. Miller was the first to publish the essential results on the commutator subgroup in [52]. However, he does not attach the label “commutator” to Dedekind’s correction factor F . The headline of the section in which he deals with commutators is simply “On the operation sts−1t−1”. Miller’s motivation in [52] for using the commutator concept was the classification of groups of order less than 48 up to isomorphism. In his two later publications addressing commutators, namely [53] and [54], he uses the label commutator and attributes it to Dedekind. In [53] Miller further expands the basic properties of the commutator subgroup, and he introduces the derived series of a group. He also shows that the derived series is finite and ends with the identity if and only if G is solvable. In his 1899 paper [54] Miller deals with commutators as objects that are of interest in their own right. He first develops a formula that shows that under certain conditions the product of two commutators is again a commutator. In modern notation, he is showing that [tb, a][a, b] = [tb, ab] for a, b, t in a group G. With the help of this identity he shows in Theorem I that every element of the alternating group on n letters, n ≥ 5, is a commutator, a result rediscovered over 50 years later by Ito [37] and Ore [57]. In Theorem II, Miller shows that in the holomorph of a cyclic group Cn the commutator subgroup consists entirely of commutators and is equal to Cn, if n is odd, and equal to the subgroup of index 2 in Cn in case n is even. This foreshadows later results by Macdonald [45] and others who investigate groups with cyclic commutator subgroup in which not all elements of the commutator subgroup are commutators. In his publications Miller never addresses the central issue in our context of whether the commutator subgroup always consists entirely of commutators. In [53] he states that for generating the commutator subgroup not all commutators are needed, and he says that a rather small portion will suffice for this purpose. On the other hand, he shows in Theorem I and II of [54] that for certain groups the set of commutators is equal to the commutator subgroup. However, as we will see when discussing [15] below, this question can not have been far from his mind. The first explicit statement of this question is found in Weber’s 1899 textbook [74], which is the first textbook to introduce commutators and the commutator subgroup. After referring to Dedekind’s definition of commutators, Weber states that the set of commutators is not necessarily a subgroup, but does not provide Kappe, Morse: On Commutators in groups 4 an example to prove his claim. He does prove that the commutator subgroup is generated by the set of commutators and this subgroup forms a normal subgroup. It is Fite [15] who provides the first such example in his paper “On metabelian groups”. It should be mentioned that the metabelian groups in the title are what we now call groups of nilpotency class two, or, as Fite states it, a group with an abelian group of inner automorphisms. Fite constructs a group G of order 1024 and nilpotency class 2 in which not all elements of the commutator subgroup are commutators. He attributes this example to G. A. Miller. In addition he provides a homomorphic image of G that has order 256, in which the set of commutators is not equal to the commutator subgroup. We discuss this in detail in Section 5. To conclude our early history of commutators we mention a 1903 paper by Burnside [5]. As detailed earlier, commutators arose out of the development of group characters. Burnside uses characters to obtain a criterion for when an element of the commutator subgroup is the product of two or more commutators. So we have come full circle! This criterion was later extended by Gallagher [17]. We discuss this in detail in Section 6. There seemed to be little interest in the topic of commutators for the 30 years following 1903. It should be kept in mind that the familiar notation for commutators had not yet been developed and its absence apparently stifled further development. The first occurrence of the commutator notation we could find is in Levi and van der Waerden’s seminal paper on the Burnside groups of exponent 3 [41]. They denote the commutator of two group elements i, j as (i, j) = iji−1j−1 and make creative use of this notation in their proofs. The first textbook using the new notation is by Zassenhaus [76]. There he gives familiar commutator identities, for example, the expansion formulas for products, but not the Jacobi identity. However, Zassenhaus states that in a group with abelian commutator subgroup the following “strange” (merkwürdig) rule holds: (a, b, c)(b, c, a)(c, a, b) = e. As the source for the definitions, notation and formulas in his section on commutators, Zassenhaus refers to Philip Hall’s paper [31], which appeared after [41]. The new notation made it possible to develop a commutator calculus to solve a variety of group theoretic problems that had not been previously accessible. In turn, the extended use of commutators as a tool brought about renewed interest in questions about commutators themselves, in particular the question on when the set of commutators is a subgroup. The remainder of the paper focuses on this question. There are significant topics about commutators that we do not cover in this paper. These topics include: viewing the commutator operation as a binary operation; Levi’s characterization of groups in which the commutator operation is associative [40]; conditions for when a product of commutators is guaranteed to be a commutator [34]; and investigations into an axiomatic treatment of the commutator laws by Macdonald and Neumann ([49], [48], and [50]) and by Ellis [14]. In the following two sections we give a survey of conditions which imply that either the set of commutators is equal to the commutator subgroup or unequal to it. Sufficient conditions for equality are rather scarce and not very powerful. As Macdonald acknowledges in [43], a forerunner of [47], there are fundamental logical Kappe, Morse: On Commutators in groups 5 difficulties in this area, for example, the main theorem of [2] implies that there is no effective algorithm for deciding whether an element is a commutator when G is a finitely presented group. However, there are necessary and sufficient conditions for an element of a finite group to be a commutator using the irreducible characters of the group. Hence, from the character table of a finite group we can read off if every element of the commutator subgroup is a commutator. Details of this can be found in Section 6. We introduce the following notation to facilitate our discussion. For a group G let K(G) = {[g, h] | g, h ∈ G} be the set of commutators of G and set G′ = 〈K(G)〉, the commutator subgroup of G. We say that the group element g is a commutator if it is an element of K(G) and a noncommutator otherwise. In Section 4 we construct various minimal examples of groups such that the commutator subgroup contains a noncommutator. These examples are minimal with respect to the order of the group G and the order of G′, respectively. With the help of GAP [19] we construct minimal examples G and H where G is a perfect group G such that K(G) 6= G′ and H is a group in which H ′ ∩ Z(H) is generated by noncommutators. As mentioned earlier, the first examples of groups with the set of commutators not equal to the commutator subgroup are finite nilpotent 2-groups of class 2. In Section 5 we develop a general construction for nilpotent p-groups of class 2 such that the commutator subgroup contains a noncommutator. This construction is obtained by finding various covering groups à of an elementary abelian p-group A of rank n ≥ 4. By a counting argument it is always the case that K(Ã) 6= à . We look at homomorphic images of two covering groups resulting in groups of order p8 with exponent p and p2, respectively, such that the set of commutators is unequal to the commutator subgroup. These groups appear in the literature ([47] and [67]) and various ad-hoc methods are used to show that the commutator subgroup contains a noncommutator. The question arises: What is the smallest integer n such that for a given prime p there exists a group G of order pn with G′ 6= K(G)? We conclude Section 5 with an answer to this question. For a group G the function λ(G) denotes the smallest integer n such that every element of G′ is a product of n commutators. This function was introduced by Guralnick in his dissertation [23]. The statement K(G) 6= G′ is then equivalent to λ(G) > 1. In Section 6 we consider conditions for upper and lower bounds for λ(G), as well as provide conditions and examples when λ(G) can be specified exactly. Some of these results involve character theory, in particular, to provide a necessary and sufficient condition on a finite group G such that λ(G) = n. In this section we include a well known example by Cassidy [8]. This is a group of nilpotency class 2 and it is claimed there is no bound on the number of commutators in the product representing an element of the commutator subgroup. However, a typographical error impacts the verification of this claim made in [8]. We include a slightly more general proof of the claim. For most problems one encounters in group theory the solution in the cyclic case is trivial. Not so here, where the situation for cyclic commutator subgroups is a microcosm for the complexity of the general case. In Section 7 we survey groups Kappe, Morse: On Commutators in groups 6 with cyclic commutator subgroup in which the commutator subgroup contains a noncommutator. Many results in Sections 2 through 7 have been extended to higher terms of the lower central series. A survey of these results is the topic of Section 8. The topic of the final section is a report on the current status on what has been called in the literature the Ore Conjecture (see [1], [4], [13], [22], [72] and [73]), which states that every element in a nonabelian finite simple group is a commutator. The Ore Conjecture is still open for some of the finite simple groups of Lie type over small fields. The details are given in a table at the end of the paper. There are many contributions on the Ore Conjecture in the literature concerning various types of semisimple and infinite simple groups (see for example [58] and [59]). These contributions go beyond the scope of this survey. 2 Conditions for equality In this section we discuss mostly conditions implying that the set of commutators is equal to the commutator subgroup. There are two types of such conditions. Those of the first type are conditions on the structure of the group or the commutator subgroup that allow us to conclude the commutator subgroup contains only commutators. Those of the second type are restrictions on the order of a group or its commutator subgroup. These restrictions are mainly derived from the structural conditions of the first type. Showing that the restrictions on the orders are best possible leads to the minimal examples discussed in Sections 4 and 5. We start with conditions on the structure of the group. One of the most versatile results is due to Spiegel. Theorem 2.1 ([68]) Suppose the group G contains a normal abelian subgroup A with cyclic factor group G/A. Then K(G) = G′. Motivated by results in [44], Liebeck in [42] gives a necessary and sufficient condition that an element of the commutator subgroup is a commutator provided the group has nilpotency class 2. Using this condition, he shows that for a group G it follows K(G) = G′ whenever G′ ⊆ Z(G) and d(G′) ≤ 2, where d(G′) denotes the minimal number of generators of G′, and he gives an example that this cannot be extended to rank 4 or greater. Rodney in [62] extends these results. In particular, he shows the following. Theorem 2.2 ([62]) The following two conditions on a group G imply G′ = K(G): (i) G is nilpotent of class two and the minimal number of generators of G′ does not exceed three; (ii) G′ is elementary abelian of order p3. The following result by Guralnick generalizes one of Rodney in [62]. Kappe, Morse: On Commutators in groups 7 Theorem 2.3 ([28]) Let P be a Sylow p-subgroup of G with P ∗ = P ∩G′ abelian and d(P ∗) ≤ 2. Then P ∗ ⊆ K(G). Similarly, Guralnick obtains the following result if p > 3. Theorem 2.4 ([28]) IfG′ is an abelian p-subgroup ofG with p > 3 and d(G′) ≤ 3, then G′ = K(G). For nilpotent groups, in particular for finite p-groups, the following conditions of the first type are useful results for arriving at sufficient conditions of the second type. Theorem 2.5 ([61]) If G is nilpotent and G′ is cyclic, then G′ = K(G). Theorem 2.6 ([39]) Let G be a finite p-group with G′ elementary abelian of rank less than or equal to three. Then K(G) = G′. With the exception of some additional conditions of type one (in the case of cyclic commutator subgroups) that we will consider in a later section, Theorems 2.1 – 2.6 are the tools currently available for arriving at sufficient conditions on the orders of G and G′ that imply K(G) = G′. We start with sufficient conditions on the orders of G and G′, which are shown to be best possible in Section 4. Theorem 2.7 ([26]) Let G be a group. If (i) G′ is abelian and |G| < 128 or |G′| < 16 or (ii) G′ is nonabelian and |G| < 96 or |G′| < 24, then K(G) = G′. Many examples of groups whose commutator subgroup contains a noncommutator are groups of prime power order. The question arises: For a p-group G of order pn, what is the largest n such that we can guarantee that K(G) = G′? As we show in Section 5, the following result is best possible. Theorem 2.8 ([39]) Let p be a prime and G a group of order pn. Then G′ = K(G) if n ≤ 5 for odd p and n ≤ 6 for p = 2. 3 Conditions for inequality In this section we discuss conditions that lead to the conclusion that the set of commutators is not equal to the commutator subgroup. However, in some cases restrictions are imposed on the structure of the group or the commutator subgroup, and then the conditions for inequality turn out to be necessary and sufficient under these restrictions. These sufficient conditions often lead to the construction of families of groups in which the commutator subgroup always contains a noncommutator. Often the objective is to find minimal examples in a certain class of groups. The conditions discussed in this section come mainly from [47], [36], [29], and [39]. The selection is based on their relevance in the next two sections, which includes the discussion of minimal examples. Kappe, Morse: On Commutators in groups 8 We start with an almost obvious criterion that one obtains by comparing the number of possible distinct commutators with the number of elements in the commutator subgroup. The condition is stated formally for the first time in [47], but earlier applications can be found in [44] and [18]. Theorem 3.1 ([47]) If G is any group and if |G : Z(G)|2 < |G′|, then there are elements in G′ that are not commutators. As Macdonald observes, the criterion is very well suited for groups with central commutator subgroups. With the help of Theorem 3.1, Macdonald constructs a large family of groups of nilpotency class 2 with the property that the set of commutators is not equal to the commutator subgroup. Isaacs’ motivation in [36] for stating his criterion is similar to Macdonald’s. He says that it is well known for a group G that not every element of G′ need be a commutator, but what is less well known is a convenient source of finite groups that are examples of this phenomenon. His examples are wreath products satisfying the following criterion. Theorem 3.2 ([36]) Let U and H be finite groups with U abelian and H nonabelian. Let G = U oH be the wreath product of U and H. Then G′ contains a noncommutator if ∑ A∈A ( 1 |U | )[H:A] ≤ 1 |U | , where A is the set of maximal abelian subgroups of H. In particular, this inequality holds whenever |U | ≥ |A|. The above construction yields both solvable and nonsolvable groups with the set of commutators not equal to the commutator subgroup. Choosing H simple and U large enough leads to perfect groups, that is, groups such that G′ = G, with the desired property. Guralnick’s goal in [29] is to determine bounds on a group G and its commutator subgroup G′ such that G′ = K(G) always holds whenever the respective orders are below these bounds. The following criterion rules out groups with a “large” abelian commutator subgroup. Theorem 3.3 ([29]) Suppose that A is an abelian group of even order. Then there exists a group G with G′ ∼= A and G′ 6= K(G) if and only if A ∼= C2 × A1 × A2 × A3 or A ∼= C2α × A1 × A2, where the Ai are nontrivial abelian groups and α ≥ 2. In [39] sufficient conditions on the nilpotency class and certain elements belonging to the center of a group G are established that guarantee that K(G) 6= G′. This leads to three classes of groups with the property that the commutator subgroup contains a noncommutator. The groups of smallest order in these classes are finite p-groups and appear as minimal examples in Section 5. As it turns out, we need to establish different criteria depending on whether p > 3, p = 3, or p = 2, respectively, which form the three classes of groups. Kappe, Morse: On Commutators in groups 9 Proposition 3.4 ([39]) Let p ≥ 5 be a prime and H = 〈a, b〉 be a nilpotent group of class exactly 4 with [b, a, b] ∈ Z(H) and exp(H ′) = p. Then K(H) 6= H ′. Proposition 3.5 ([39]) Let H = 〈a, b〉 be a nilpotent group of class exactly 4 with a3, b9, [b, a, b] ∈ Z(H). Then K(H) 6= H ′. Proposition 3.6 ([39]) Let H = 〈a, b, c〉 be a group of class 3 precisely. If a4, b2, c2, [a, c], [b, c] and (ab)2 ∈ Z(H), then K(H) 6= H ′. 4 Some minimal examples MacHale in [51] lists 47 conjectures about groups that are known to be false and asks for a minimal counterexample for each. Conjecture 7 in his paper states “In any group G, the set of all commutators forms a subgroup”. MacHale indicates that a minimal counterexample is known. In fact this is the topic of Guralnick’s Ph.D. dissertation [23] and several subsequent papers, in particular [26] and [28]. In [26] examples are constructed or cited to show that the conditions of Theorem 2.7 are tight. In this section and subsequent sections we consider the following classes of groups and find groups of minimal order in each: Groups such that the commutator subgroup is not equal to the set of commutators and (i) the commutator subgroup is abelian; (ii) the commutator subgroup is abelian of order 16; (iii) the commutator subgroup is nonabelian; (iv) the commutator subgroup is nonabelian of order 24; (v) the intersection of the commutator subgroup and the center is generated by noncommutators; (vi) the group is perfect. In Section 5 we construct a metabelian group G of order 27 such that G′ has order 16 and K(G) 6= G′. This group is of minimal order in classes (i) and (ii). It follows from Theorem 2.6 that G′ is not cyclic and in fact in any minimal example H in classes (i) and (ii) the commutator subgroup H ′ can not be cyclic. Hence we consider the following variant of classes (i) and (ii): Groups such that the commutator subgroup is not equal to the set of commutators and (i’) the commutator subgroup is cyclic; (ii’) the commutator subgroup is cyclic of order 60. Kappe, Morse: On Commutators in groups 10 The groupG constructed in Example 7.10 has order 240 such that its commutator subgroup is cyclic of order 60 and K(G) 6= G′. This group is of minimal order in classes (i’) and (ii’). Of course MacHale was interested in the smallest group order such that the commutator subgroup contains a noncommutator. As we see below, such groups are in class (iii). Minimal examples for each class (i) – (vi), (i’) and (ii’) can be found using GAP by searching its small groups library. For example, Rotman [65] states that via computer search the smallest examples G such that K(G) 6= G′ have order 96. Rotman was apparently unaware of Guralnick’s earlier work. The following example gives explicit constructions of the two nonisomorphic groups of order 96 whose commutator subgroups contain noncommutators. Example 4.1 ([23]) There are exactly two nonisomorphic groups G of order 96 such that K(G) 6= G′. In both cases G′ is nonabelian of order 32 and |K(G)| = 29. (a) Let G = H o 〈y〉, where H = 〈a〉 × 〈b〉 × 〈i, j〉 ∼= C2 ×C2 ×Q8 and 〈y〉 ∼= C3. Let y act on H as follows: ay = b, by = ab, iy = j and jy = ij. (b) Let H = N o 〈c〉, where N = 〈a〉 × 〈b〉 ∼= C2 ×C4 and 〈c〉 ∼= C4. Let c act on N by ac = a and bc = ab. Let G = H o 〈γ〉 with 〈γ〉 ∼= C3, where aγ = c2b2, bγ = cba, cγ = ba. The group of Example 4.1 (a) appears in Dummit and Foote’s textbook [12] as an example of a group G with K(G) 6= G. No claim of minimality is made. Guralnick in [26] describes the following class of groups that yields many examples of groups G in which K(G) 6= G′. Let G1 = 〈a, b, x |a = b = x = 1, xax−1 = b, xbx−1 = ab, (4.1.1) a = b, aba−1 = b−1〉. Then G1 = 〈H1, x〉, where H1 = G1 = 〈a, b〉 ∼= Q8. Choose G2 to be any nonabelian group with normal abelian subgroup H2 of index 3. Then there exists y ∈ G2 such that G2 = 〈H2, y〉. Let G be the subgroup of G1 × G2 generated by H1 ×H2 and the element (x, y). Then G has order 24|H2| and K(G) 6= G′. Note that G′ = G1 ×G2. In particular, for any 1 6= g ∈ G2 the element (a2, g) is not a commutator. Our next three examples arise from this construction for particular choices of G2. Example 4.2 Let G1 be defined as in (4.1.1) and take G2 = A4 and H2 = G2 ∼= C2 × C2. Then G has order 96 and G′ ∼= Q8 × C2 × C2. The group constructed in Example 4.2 is isomorphic to the group in Example 4.1 (a). However, Example 4.1 (b) cannot be constructed in this manner, since there is not another nonabelian group of order 12 with a normal subgroup of order 4. The following example gives a group of minimal order in class (iv). Kappe, Morse: On Commutators in groups 11 Example 4.3 Let G1 be the group defined in (4.1.1). Take G2 to be any nonabelian group of order 27 so that |H2| = 9 and G2 ∼= C3. Then G′ ∼= Q8 × C3, which is nonabelian of order 24. The order of G is 24 · 9 = 216. We finish this section by giving minimal examples of groups G such that K(G) 6= G′ with some additional property. The first example is a groupG in whichG′∩Z(G) is generated by noncommutators. The question of whether such a group exists was asked by R. Oliver and an example was given by Caranti and Scopolla [6]. Their example has order p14, where p is an odd prime, but it is noted in the paper that a smaller example exists. Our example with this property has order 216 which by a search of the small groups library in GAP is the smallest such example. Example 4.4 Set G1 = 〈H1, x〉 as in (4.1.1) and set G2 = 〈c, y | c = y = 1, y−1cy = c〉 ∼= C9 o C3 where H2 = C9 = 〈c〉 and C3 = 〈y〉. Let G = 〈H1 ×H2, {(x, y)}〉. Then Z(G) = 〈(a2, c3)〉 ∼= C6. However, since c3 ∈ G2, we have that (a2, c3) is a noncommutator, as needed. To find a perfect group G in which K(G) 6= G′ we can use the group construction and criterion from Theorem 3.2 due to Isaacs [36]. The smallest perfect group obtainable from this construction is the following. Example 4.5 Let G = C2 o A5. Then G is a perfect group with |G| = 260 · 60. To see that G′ 6= K(G) we note that the maximal abelian subgroups of A5 are its Sylow subgroups. There are ten maximal abelian subgroups of order 3, five maximal abelian subgroups of order 4, and six maximal abelian subgroups of order 5. This gives 10 ( 1 3 )20 + 5 ( 1 4 )15 + 6 ( 1 5 )12 ≤ 1 60 , which by Theorem 3.2 shows that K(G) 6= G′. A search of the perfect groups in GAP shows that the smallest perfect group G that contains an element that is not a commutator has order 960. This group can be visualized in the following way. Let H = C5 2 oA5, where we think of A5 having a “wreath action” on C5 2 . Set G = H/Z(H). Then G is a perfect group of order 25 · 60 · 12 and has the property that K(G) 6= G ′.