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Perfect group

In group theory, a perfect group is a group G that equals its own derived subgroup [G, G], meaning every element of G can be expressed as a product of commutators [g, h] = g h g^{-1} h^{-1} for g, h \in G.^[1]^[2]^[3] Equivalently, the abelianization G^{\mathrm{ab}} = G / [G, G] of a perfect group is the trivial group, implying that G admits no nontrivial homomorphisms to abelian groups.^[1]^[2] Perfect groups play a central role in the structure theory of finite groups, particularly in the classification of finite simple groups, as every non-abelian simple group is perfect.^[3] This property is preserved under quotients, so any quotient of a perfect group is also perfect, and direct products of perfect groups are perfect.^[2]^[1] However, perfectness is not inherited by subgroups; for instance, the trivial subgroup of any group is perfect, but nontrivial proper subgroups of perfect groups need not be.^[1] Notable examples include the alternating group A_5 of order 60, which is the smallest nontrivial perfect group and the smallest non-abelian simple group.^[2] Other examples are the special linear groups \mathrm{SL}_n(\mathbb{F}_q) for n \geq 2 and finite fields \mathbb{F}_q with q > 3, which are perfect but not always simple due to possible nontrivial centers.^[2]^[3] The binary icosahedral group of order 120 is the unique finite perfect group of that order up to isomorphism.^[2] In contrast, all solvable groups except the trivial group are imperfect, as their derived series terminates at the trivial subgroup.^[1]

Fundamentals

Definition

In group theory, a perfect group is defined as a group G such that G = [G, G], where [G, G] denotes the commutator subgroup of G, generated by all commutators of the form [g, h] = g h g^{-1} h^{-1} for g, h \in G. The commutator subgroup [G, G] is the smallest normal subgroup of G whose quotient is abelian, known as the abelianization of G.^[4] Consequently, a perfect group has trivial abelianization, implying it admits no nontrivial abelian quotients.^[5] The concept of perfect groups arose in the work of Issai Schur on projective representations of finite groups.^[6] Equivalent characterizations include the condition that every element of a perfect group G is a product of commutators.^[7] Additionally, G is perfect if and only if its first cohomology group H^1(G, \mathbb{C}^*) vanishes, where \mathbb{C}^* carries the trivial G-action.

Basic properties

A perfect group G has the property that any quotient G/N by a normal subgroup N is also perfect, since the commutator subgroup of the quotient is the image of [G, G] = G under the canonical projection.^[1] Nontrivial perfect groups are not solvable, as the derived series of a solvable group eventually reaches the trivial subgroup, whereas for a perfect group, the first derived subgroup G' = [G, G] = G, so the series remains constant at G.^[1] The center Z(G) of any group G satisfies [G, Z(G)] = 1, since elements of the center commute with all elements of G. For perfect groups, Z(G) need not be trivial; for instance, \mathrm{SL}(2, 5) is a perfect group of order 120 with center of order 2.^[8] For finite perfect groups, the order |G| is divisible by 4. If |G| is not divisible by 8, then it must be divisible by 3; this follows from Sylow theory, as the Sylow 2-subgroup cannot be cyclic (which would yield a non-trivial abelian quotient, contradicting perfectness), and representation-theoretic considerations rule out certain small 2-subgroup structures without a factor of 3. The derived series of a perfect group terminates immediately at G itself, since G^{(k)} = G for all k \geq 1, in contrast to the terminating series of solvable or nilpotent groups that reach the trivial group.^[1] Every perfect group G admits a universal central extension E, which is a central extension $1 \to K \to E \to G \to 1 with kernel K = H_2(G, \mathbb{Z}) (the Schur multiplier) such that E is perfect and any other central extension of G factors uniquely through E.^[9]

Examples

Finite perfect groups

The smallest non-abelian perfect group is the alternating group A_5, which has order 60. This group is simple, and its perfection follows from the fact that it equals its own commutator subgroup. All finite non-abelian simple groups are perfect. For such a group G, the commutator subgroup G' is normal in G; since G is non-abelian, G' > 1, and by simplicity, G' = G. Examples include the projective special linear groups \mathrm{[PSL](/page/PSL)}(n, q) for n \geq 2 and prime power q \geq 2, excluding the cases \mathrm{[PSL](/page/PSL)}(2,2) \cong S_3 and \mathrm{[PSL](/page/PSL)}(2,3) \cong A_4, which are solvable. Quasisimple groups provide non-simple examples of finite perfect groups. These are perfect central extensions of non-abelian simple groups, meaning the quotient by their center is simple non-abelian.^[10] A representative example is the special linear group \mathrm{SL}(2,5), which has order 120 and center of order 2; its quotient is \mathrm{[PSL](/page/PSL)}(2,5) \cong A_5..html) Direct products of non-abelian simple groups yield further non-simple perfect groups. The commutator subgroup of a direct product G \times H is G' \times H', so if G and H are non-abelian simple (hence perfect), then G \times H is perfect.^[11] For instance, A_5 \times A_5 has order 3600 and is perfect but not simple, as it has normal subgroups isomorphic to A_5.^[11] Finite Steinberg groups \mathrm{St}(n, R), defined for integer n \geq 3 and commutative ring R with no element of additive order 2, are also perfect.^[12] These groups arise as universal central extensions of the elementary subgroup E(n, R) of \mathrm{GL}(n, R), and their perfection holds under the specified conditions on n and R.^[13] By the classification of finite simple groups, every finite perfect group is a central product of quasisimple groups, each of which is a central extension of a non-abelian simple group; thus, all finite perfect groups are constructed from the finite simple groups via such extensions and products.^[10]

Infinite perfect groups

Infinite perfect groups arise in both continuous and discrete settings, providing rich examples beyond their finite counterparts. In the realm of Lie groups, the special linear group \mathrm{SL}(n, \mathbb{R}) is perfect for all n \geq 2. This follows from the fact that its commutator subgroup equals itself, except in the finite exceptional cases \mathrm{SL}(2,2) and \mathrm{SL}(2,3), which are irrelevant here. Similarly, \mathrm{SL}(n, \mathbb{C}) shares this property for n \geq 2. In contrast, the general linear group \mathrm{GL}(n, \mathbb{R}) is not perfect for n \geq 2, as its derived subgroup is precisely \mathrm{SL}(n, \mathbb{R}), yielding a non-trivial abelian quotient isomorphic to \mathbb{R}^\times / \{\pm 1\}.^[14] Topological aspects are prominent in non-compact simple Lie groups, which are perfect by virtue of their simplicity. For instance, the projective special linear groups \mathrm{[PSL](/page/PSL)}(n, \mathbb{R}) and \mathrm{[PSL](/page/PSL)}(n, \mathbb{C}) are simple for n \geq 2, hence perfect, as non-abelian simple groups admit no non-trivial abelian quotients. These groups exemplify perfect Lie groups of non-compact type, where the Lie algebra's simplicity ensures the group's derived subgroup coincides with itself.^[15] Discrete infinite perfect groups include arithmetic examples like \mathrm{SL}(3, \mathbb{Z}), which is finitely presented and perfect. More generally, free products of non-abelian simple groups yield infinite perfect groups; the abelianization of such a free product is the direct product of the individual abelianizations, which are trivial for perfect factors. A concrete case is the free product A_5 * A_5, where A_5 is the alternating group on five letters, producing an infinite perfect group with a free subgroup of finite index.^[16]^[17] Constructions via amalgamated free products and HNN extensions also generate infinite perfect groups. For example, certain amalgamations over finite subgroups of simple groups preserve perfection when the overall abelianization vanishes. HNN extensions of perfect groups, under conditions ensuring the stable letter does not introduce abelian quotients, similarly yield perfect examples, as seen in presentations of groups like \mathrm{SL}(3, \mathbb{Z}). These methods highlight the flexibility in building infinite perfect structures from simpler building blocks.^[16]

Key theorems

Ore's conjecture

In 1951, Øystein Ore conjectured that every element of every finite non-abelian simple group is a commutator. This statement posits that for any such group G and any g \in G, there exist x, y \in G such that g = [x, y] = x y x^{-1} y^{-1}. Ore himself established the conjecture for alternating groups A_n with n \geq 5. Partial progress followed in subsequent decades. In the 1960s, R. C. Thompson proved the conjecture for projective special linear groups \mathrm{PSL}_n(q).^[18] During the 1980s, S. A. Gow verified it for symplectic groups \mathrm{PSp}_{2n}(q) where q \equiv 1 \pmod{4}. Further advancements in the 1990s and early 2000s covered exceptional groups of low rank and most groups of Lie type over fields with at least eight elements, as shown by Ellers and Gordeev in 2002.^[18] The conjecture was fully resolved affirmatively in 2010 by Martin Liebeck, Eamonn O'Brien, Aner Shalev, and Pham Tiep, who employed a combination of character-theoretic methods, inductive arguments on group structure, and extensive computational verification using the Classification of Finite Simple Groups (CFSG).^[19] Their proof leverages estimates on character values and the Frobenius formula to bound the number of solutions to commutator equations, incorporating probabilistic techniques for handling large character sums in groups of Lie type.^[18] Sporadic simple groups were treated separately via case-by-case analysis.^[19] This resolution confirms that every finite non-abelian simple group is generated by its commutators, directly reinforcing the fact that such groups are perfect, as their derived subgroup equals the entire group.^[19] The result extends naturally to quasisimple groups, where every element outside specific small exceptions (like $3 \cdot A_6) is a commutator, and has implications for centralizers in finite classical groups.^[18] It also applies to almost simple groups, providing bounds on commutator widths and advancing broader questions in finite group generation.^[19]

Grün's lemma

Grün's lemma asserts that if G is a perfect group, then the center of the quotient group G/Z(G) is trivial, where Z(G) denotes the center of G.^[20] This result, originally proved by Otto Grün in 1935 as part of his foundational contributions to the structure of finite groups, highlights a key distinction between perfect groups and other classes, such as nilpotent groups where central series behave differently.^[21] The proof proceeds using the three-subgroups lemma, a standard identity in commutator calculus stating that if [H,K,L] \leq N and [K,L,H] \leq N for subgroups H,K,L of G and a normal subgroup N, then [L,H,K] \leq N. Since G is perfect, [G,G] = G, and elements of Z(G) commute with all of G, so [G, Z(G)] = 1. To establish the centerlessness of G/Z(G), consider the preimage \zeta_2(G) of Z(G/Z(G)) under the natural projection; \zeta_2(G) lies in the second term of the upper central series, and [G, \zeta_2(G)] \leq Z(G). Since elements of [G, \zeta_2(G)] are central, the commutator map satisfies the property that [G, \zeta_2(G)] = [[G,G], \zeta_2(G)]. Now, [G, \zeta_2(G), G] = [[G, \zeta_2(G)], G] \leq [Z(G), G] = 1 and [\zeta_2(G), G, G] = [[\zeta_2(G), G], G] \leq 1. Applying the three-subgroups lemma with H = G, K = \zeta_2(G), L = G yields [G, G, \zeta_2(G)] \leq 1. Thus, [G, \zeta_2(G)] = 1, implying \zeta_2(G) \subseteq Z(G). Therefore, Z(G/Z(G)) = 1.^[22] This lemma has significant applications in classifying centerless perfect groups, as it characterizes them precisely as quotients of arbitrary perfect groups by their centers, aiding the study of their holomorphs and structural properties.^[23] It is also instrumental in investigating Schur multipliers of perfect groups, where the triviality of the center in such quotients informs the detection of non-trivial central extensions and the computation of H_2(G, \mathbb{Z}).^[22] Extensions of Grün's lemma apply to higher commutators and the solvable radical, showing that the upper central series of a perfect group stabilizes after the first non-trivial term, with \zeta_k(G) = Z(G) for all k \geq 2; this follows directly from the centerlessness of G/Z(G).^[20] Historically, the lemma emerged amid early 20th-century advancements in understanding nilpotent and perfect group structures, influenced by works on commutator subgroups and central extensions during the interwar period.^[21]

Variants

Quasi-perfect groups

A group G is quasi-perfect if its commutator subgroup G' is perfect, that is, G' = [G', G'].^[24] Equivalently, in derived series notation, G^{(1)} = G^{(2)}.^[24] Quasi-perfect groups have a perfect derived subgroup, and thus their abelianization G/G' is abelian.^[24] They arise prominently in algebraic K-theory, where the plus construction applies to their classifying spaces when the fundamental group is quasi-perfect.^[25] Finite examples include the Steinberg groups \mathrm{St}(R; \mathbb{Z}/m\mathbb{Z}) for a finite unital ring R (such as a finite field) and integer m \geq 2; these are quasi-perfect with commutator subgroup isomorphic to the perfect group \mathrm{St}_m(R).^[26] Infinite examples include central extensions of perfect groups by cyclic groups, such as \mathrm{St}(R; \mathbb{Z}/m\mathbb{Z}) for infinite unital rings R (e.g., R = \mathbb{Z}) and m \geq 2, where the commutator subgroup is the perfect elementary subgroup E(R).^[26] Every perfect group is quasi-perfect, as G = G' implies G' = [G', G'].^[24] The converse fails; for instance, the Steinberg group \mathrm{St}(R; \mathbb{Z}/m\mathbb{Z}) for m > 1 is typically a non-trivial central extension and hence not perfect.^[26]

Superperfect groups

A perfect group G is called superperfect if its second integral homology group vanishes, that is, H_2(G, \mathbb{Z}) = 0.^[27] This condition is equivalent to the Schur multiplier M(G) of G being trivial.^[28] Superperfect groups admit no non-trivial central extensions, meaning that any central extension of a superperfect group G by an abelian group splits.^[27] Since superperfect groups are perfect by definition, this homological vanishing strengthens the derived subgroup condition. In particular, simple superperfect groups are centerless.^[28] Examples of superperfect groups include \mathrm{PSL}(3,3) and \mathrm{PSU}(3,3), as are sporadic simple groups like the Mathieu group M_{11}.^[29] For \mathrm{PSL}(2,q), the Schur multiplier is trivial when q is a power of 2 greater than 4, so \mathrm{PSL}(2,q) is superperfect in those cases.^[28] One construction of superperfect groups involves stem extensions of centerless perfect groups, where the extension kernel is contained in both the derived subgroup and the center of the covering group, but the triviality of the Schur multiplier ensures such groups coincide with their universal central extensions.^[27] The notion of superperfect groups emerged in the context of studying projective representations and Schur multipliers during the 1960s, building on Schur's foundational work and contributing to efforts in classifying finite simple groups and their covers.^[28]

Connections to homology

Implications for group homology

A fundamental homological characterization of perfect groups arises in the first homology group with integer coefficients. For any group G, the first homology group is isomorphic to the abelianization: H_1(G, \mathbb{Z}) \cong G/[G,G].^[30] Thus, a group G is perfect if and only if H_1(G, \mathbb{Z}) = 0, reflecting the triviality of its abelianization.^[30] This connection extends to higher dimensions through variants like superperfect groups, which are perfect groups with the additional property that H_2(G, \mathbb{Z}) = 0.^[31] For perfect groups, the second homology group H_2(G, \mathbb{Z}) classifies central extensions of G, and in particular, it parametrizes the universal central extension of G, a maximal stem extension that captures all central extensions via the Schur multiplier.^[32] These homological properties have significant applications in algebraic K-theory, where universal central extensions of perfect groups like \mathrm{SL}_n(k) yield the Steinberg groups, whose kernels define the higher K-groups K_2(n,k) and ultimately K_2(k) via Matsumoto's theorem.^[33] Moreover, the vanishing of low-dimensional homology groups for perfect groups provides key topological invariants, as H_*(G, \mathbb{Z}) computes the homology of the classifying space BG, linking perfection to the homotopy type of associated spaces.^[32] A representative example is the alternating group A_5, which is perfect and thus has H_1(A_5, \mathbb{Z}) = 0, but exhibits non-trivial higher homology with H_2(A_5, \mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}, corresponding to its universal central extension by the binary icosahedral group.^[32]

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