# Commutator subgroup

In group theory, the **commutator subgroup** (also called the **derived subgroup**) of a group \( G \) is the subgroup generated by all commutators \( [a,b] = aba^{-1}b^{-1} \) of elements of \( G \). It is usually denoted \( [G,G] \), \( G' \), or \( \Gamma_2(G) \), and it measures the failure of \( G \) to be abelian: it is trivial exactly when \( G \) is commutative, and it is the unique smallest normal subgroup whose quotient is abelian.<sup>[1](https://encyclopediaofmath.org/wiki/Commutator_subgroup)</sup><sup> • </sup><sup>[2](https://mathworld.wolfram.com/CommutatorSubgroup.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( [G,G] = \langle [a,b] \mid a,b \in G \rangle \), the subgroup generated by all commutators; also called the derived subgroup<sup>[1](https://encyclopediaofmath.org/wiki/Commutator_subgroup)</sup> |
| Universal property | The unique smallest normal subgroup \( N \) of \( G \) with \( G/N \) abelian; a quotient \( G/N \) is abelian if and only if \( [G,G] \subseteq N \)<sup>[2](https://mathworld.wolfram.com/CommutatorSubgroup.html)</sup><sup> • </sup><sup>[1](https://encyclopediaofmath.org/wiki/Commutator_subgroup)</sup> |
| Normality | \( [G,G] \) is normal in \( G \), and in fact fully characteristic, invariant under every endomorphism of \( G \)<sup>[1](https://encyclopediaofmath.org/wiki/Commutator_subgroup)</sup> |
| Range | It can be as small as the trivial subgroup (abelian groups) or as large as \( G \) itself (perfect groups)<sup>[2](https://mathworld.wolfram.com/CommutatorSubgroup.html)</sup> |
| Abelianization | \( G^{\mathrm{ab}} = G/[G,G] \) is the largest abelian quotient of \( G \); in topology, \( H_1(G) = G/[G,G] \)<sup>[3](https://www.ub.edu/topologia/casacuberta/cursos/gtv1516_3.pdf)</sup> |
| Commutators vs. subgroup | The set of commutators need not be closed under products; the smallest finite group in which some element of the derived subgroup is not a commutator has order 96<sup>[4](https://ems.press/content/serial-article-files/30117)</sup> |
| Ore conjecture | Every element of a finite non-abelian simple group is a single commutator; conjectured by Øystein Ore in 1951 and proved in 2010<sup>[5](https://www.math.auckland.ac.nz/~obrien/research/ore.pdf)</sup> |

## Definition and basic properties

For elements \( a, b \) of a group \( G \), the commutator \( [a,b] = aba^{-1}b^{-1} \) records how far \( a \) and \( b \) are from commuting: it equals the identity precisely when \( ab = ba \). The commutator subgroup is

\[ [G,G] = \langle [a,b] \mid a,b \in G \rangle, \]

the subgroup generated by all such elements. The generating set is needed because the set of commutators alone is not closed under multiplication in general; every element of \( [G,G] \) is a product of commutators, but not necessarily a single commutator.<sup>[6](https://mathweb.ucsd.edu/~lbuzaglo/teaching/103AF24/103AF24L18.pdf)</sup><sup> • </sup><sup>[7](https://www.cambridge.org/core/books/abs/groups-st-andrews-2005/on-commutators-in-groups/7B968F4A30CA585ACC9D380EDBF12077)</sup>

**Normality.** The commutator subgroup is a normal subgroup of \( G \). The key identity is that conjugation of a commutator is again a product of commutators:

\[ g[a,b]g^{-1} = [ga,b]\,[b,g], \]

so \( [G,G] \) is closed under conjugation.<sup>[8](https://math.ucla.edu/~sharifi/notes/algebra-ch04.html)</sup> More strongly, it is a fully characteristic subgroup, invariant under every endomorphism of \( G \), because endomorphisms send commutators to commutators.<sup>[1](https://encyclopediaofmath.org/wiki/Commutator_subgroup)</sup>

**The universal property.** If \( N \) is a normal subgroup of \( G \), then \( G/N \) is abelian if and only if \( [G,G] \subseteq N \). This follows because the quotient is abelian exactly when every commutator lies in \( N \). Consequently \( [G,G] \) is the unique smallest normal subgroup with abelian quotient, and any subgroup containing it is normal.<sup>[2](https://mathworld.wolfram.com/CommutatorSubgroup.html)</sup><sup> • </sup><sup>[1](https://encyclopediaofmath.org/wiki/Commutator_subgroup)</sup>

## Abelianization

The quotient \( G^{\mathrm{ab}} = G/[G,G] \) is called the abelianization of \( G \), the largest quotient of \( G \) that is a commutative group.<sup>[3](https://www.ub.edu/topologia/casacuberta/cursos/gtv1516_3.pdf)</sup> It satisfies a universal property: the projection \( G \to G^{\mathrm{ab}} \) is a homomorphism onto an abelian group such that every homomorphism from \( G \) to an abelian group factors uniquely through it.<sup>[3](https://www.ub.edu/topologia/casacuberta/cursos/gtv1516_3.pdf)</sup>

In the eight-element quaternion group \( \{\pm 1, \pm i, \pm j, \pm k\} \), the commutator subgroup is \( \{\pm 1\} \), and the abelianization is a copy of \( \mathbb{Z}_2 \times \mathbb{Z}_2 \); for instance, the images of \( i \) and \( j \) commute there although \( i \) and \( j \) do not commute in \( G \).<sup>[9](https://mathworld.wolfram.com/Abelianization.html)</sup>

## Examples

- **Abelian groups:** \( [G,G] = \{e\} \), since every commutator is the identity.<sup>[8](https://math.ucla.edu/~sharifi/notes/algebra-ch04.html)</sup>
- **Dihedral groups:** in \( D_n \), computations give \( [r^i, r^j s] = r^{2i} \) and \( [r^i s, r^j s] = r^{2(i-j)} \), so \( [D_n, D_n] = \langle r^2 \rangle \), a subgroup of index 2 when \( n \) is odd and index 4 when \( n \) is even.<sup>[8](https://math.ucla.edu/~sharifi/notes/algebra-ch04.html)</sup>
- **Symmetric and alternating groups:** the commutator subgroup of the symmetric group \( S_n \) is the alternating group \( A_n \) of even permutations.<sup>[2](https://mathworld.wolfram.com/CommutatorSubgroup.html)</sup><sup> • </sup><sup>[10](https://math.berkeley.edu/~serganov/math252/Bookrep.pdf)</sup> For \( n \ge 5 \), \( A_n \) is simple, and since \( [A_n, A_n] \) is a nontrivial normal subgroup it must be all of \( A_n \).<sup>[2](https://mathworld.wolfram.com/CommutatorSubgroup.html)</sup>
- **Free groups:** the commutator subgroup \( [F_n, F_n] \) of the free group on \( n \) generators is itself a free group of infinite rank.<sup>[11](https://arxiv.org/abs/2101.05905)</sup>
- **Perfect groups:** a group equal to its own commutator subgroup is called perfect; \( A_5 \) and the binary icosahedral group \( 2I \cong SL_2(\mathbb{F}_5) \) are examples.<sup>[12](https://ncatlab.org/nlab/show/perfect%2Bgroup)</sup>

## The set of commutators and the Ore conjecture

A product of two commutators need not be a commutator, which is why the definition uses the generated subgroup. [Øystein Ore](https://www.edgepedia.org/ystein-ore) highlighted the question in his 1951 paper "Some remarks on commutators", asking when every element of the commutator subgroup is itself a commutator.<sup>[7](https://www.cambridge.org/core/books/abs/groups-st-andrews-2005/on-commutators-in-groups/7B968F4A30CA585ACC9D380EDBF12077)</sup> The smallest finite group in which some element of the derived subgroup is not a commutator has order 96.<sup>[4](https://ems.press/content/serial-article-files/30117)</sup>

Ore also conjectured in 1951 that every element of every finite non-abelian simple group is a single commutator. He established the alternating groups himself, John Thompson handled \( PSL_n(q) \), and partial results accumulated for the sporadic and Lie-type families until Martin Liebeck, Eamonn O'Brien, Aner Shalev, and Pham Huu Tiep completed the proof in 2010, using character theory together with induction and computer calculation.<sup>[5](https://www.math.auckland.ac.nz/~obrien/research/ore.pdf)</sup> The conjecture does not extend to all quasisimple groups: in \( 3.A_6 \), no element of order 12 is a commutator.<sup>[5](https://www.math.auckland.ac.nz/~obrien/research/ore.pdf)</sup>

The number of commutators needed to express an element of \( [G,G] \) is the commutator width. In the first Grigorchuk group the width is 2: every element of \( [G,G] \) is a product of two commutators, yet the group contains subgroups of infinite commutator width.<sup>[13](https://arxiv.org/abs/1710.05706)</sup>

## Derived series and solvability

Iterating the construction gives the derived series: \( G^{(0)} = G \) and \( G^{(k+1)} = [G^{(k)}, G^{(k)}] \). A group \( G \) is solvable if and only if \( G^{(s)} = 1 \) for some \( s \in \mathbb{N} \).<sup>[14](https://archive.nptel.ac.in/content/storage2/courses/111101001/modules/lec20/20.5.html)</sup> The term "solvable group" arose in [Galois theory](https://www.edgepedia.org/galois-theory) in connection with the solvability of algebraic equations by radicals, and the derived series is the most important of the subnormal series with abelian quotients used to define solvability.<sup>[15](https://encyclopediaofmath.org/wiki/Solvable_group)</sup>

## Connection with topology

Abelianization links group theory to [algebraic topology](https://www.edgepedia.org/algebraic-topology). For a path-connected topological space \( X \), the Hurewicz–Poincaré theorem identifies the first homology group with the abelianization of the fundamental group: \( H_1(X) \cong \pi_1(X, x_0)^{\mathrm{ab}} = \pi_1(X,x_0)/[\pi_1(X,x_0), \pi_1(X,x_0)] \).<sup>[3](https://www.ub.edu/topologia/casacuberta/cursos/gtv1516_3.pdf)</sup> Homology is abelian by construction, so the commutator subgroup is exactly what must be quotiented out to pass from loops to one-dimensional cycles.

## History

Commutators originated over a century ago as a by-product of computing group characters of nonabelian groups, and became objects of study in their own right soon after their introduction.<sup>[7](https://www.cambridge.org/core/books/abs/groups-st-andrews-2005/on-commutators-in-groups/7B968F4A30CA585ACC9D380EDBF12077)</sup> The name "derived subgroup" traces to [Camille Jordan](https://www.edgepedia.org/camille-jordan)'s 1870 *Traité des substitutions*, where "groupe dérivé" is used in the modern sense of the subgroup generated by the commutators.<sup>[16](https://math.stackexchange.com/questions/1331357/origin-of-the-term-derived-subgroup?noredirect=1)</sup>

## References

1. [Commutator subgroup, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Commutator_subgroup)
2. [Commutator Subgroup, Wolfram MathWorld (Weisstein, E. W., with contributions by Todd Rowland)](https://mathworld.wolfram.com/CommutatorSubgroup.html)
3. [Geometry and Topology of Manifolds 2015–2016, First Homology Group (Universitat de Barcelona course notes)](https://www.ub.edu/topologia/casacuberta/cursos/gtv1516_3.pdf)
4. [Commutator width, EMS Press](https://ems.press/content/serial-article-files/30117)
5. [Liebeck, M. W., O'Brien, E. A., Shalev, A., Tiep, P. H. (2010). The Ore conjecture. Journal of the European Mathematical Society 12(4)](https://www.math.auckland.ac.nz/~obrien/research/ore.pdf)
6. [UCSD Math 103 lecture notes: The commutator subgroup](https://mathweb.ucsd.edu/~lbuzaglo/teaching/103AF24/103AF24L18.pdf)
7. [Kappe, L.-C., Morse, R. F. (2007). On commutators in groups. Groups St Andrews 2005, Cambridge University Press, pp. 531–558](https://www.cambridge.org/core/books/abs/groups-st-andrews-2005/on-commutators-in-groups/7B968F4A30CA585ACC9D380EDBF12077)
8. [Sharifi, R. Abstract Algebra, Chapter 4: Advanced group theory, UCLA](https://math.ucla.edu/~sharifi/notes/algebra-ch04.html)
9. [Abelianization, Wolfram MathWorld](https://mathworld.wolfram.com/Abelianization.html)
10. [Serganov, M. Math 252 problem notes, UC Berkeley](https://math.berkeley.edu/~serganov/math252/Bookrep.pdf)
11. [The commutator subgroups of free groups and surface groups, arXiv:2101.05905](https://arxiv.org/abs/2101.05905)
12. [Perfect group, nLab](https://ncatlab.org/nlab/show/perfect%2Bgroup)
13. [Commutator width in the first Grigorchuk group, arXiv:1710.05706](https://arxiv.org/abs/1710.05706)
14. [NPTEL course notes: derived series and solvability](https://archive.nptel.ac.in/content/storage2/courses/111101001/modules/lec20/20.5.html)
15. [Solvable group, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Solvable_group)
16. [Origin of the term "derived subgroup", Mathematics Stack Exchange](https://math.stackexchange.com/questions/1331357/origin-of-the-term-derived-subgroup?noredirect=1)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Group theory › Group structures and subgroups*

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