Commutators $[g,h]=gh{g}^{-1}{h}^{-1}$ and the commutator subgroup $[G,G]$

Depends on

• The subgroup $\langle S \rangle$ generated by a subset, the cyclic subgroup $\langle g \rangle$, and cyclic groups
• In a group $e^{-1} = e$, $(g^{-1})^{-1} = g$ and $(gh)^{-1} = h^{-1}g^{-1}$, the order of the last product being essential

Used by

• [Sₙ,Sₙ]=Aₙ for n≥2, and [Aₙ,Aₙ]=Aₙ for n≥5 Corollary
• For an odd prime p, the p-th power map is a homomorphism on a finite group whose derived subgroup is central of exponent dividing p Corollary
• For odd p, a direct product of two Heisenberg groups is special with centre of order p², hence not extraspecial Counterexample
• Central and stem extensions Definition
• Perfect group Definition
• Quasisimple groups, components, and the layer Definition
• Special and extraspecial p-groups Definition
• Subgroup commutators and the lower central series Definition
• The abelianisation Gᵃᵇ:=G/[G,G] and its canonical map Definition
• The commutator pairing of an extraspecial p-group relative to a chosen generator of its centre Definition
• For an abelian group G, Z(G)=G and [G,G]={e} Example
• The canonical surjection from a free product to the direct product of its factors Example
• The Frattini subgroups of the dihedral and quaternion groups of order eight Example
• Commutator identities in a group whose derived subgroup is central Lemma
• Distinct minimal normal subgroups centralize one another Lemma
• For odd p, a central product of two modular groups of order p³ is a central product of a modular group with a Heisenberg group Lemma
• In a group with central derived subgroup, (xy)ⁿ=[y,x]^C(n, 2)xⁿyⁿ Lemma
• The commutator pairing is well defined on the central quotient, is bilinear over Fₚ, and is alternating Lemma
• The commutator subgroup is normal Lemma
• The Hopf-formula quotient exists Lemma
• Two elements of an extraspecial p-group with nontrivial commutator generate an extraspecial subgroup of order p³ Lemma
• An automorphism fixing the centre pointwise induces a pairing-preserving automorphism of the central quotient, with kernel the inner automorphisms Proposition
• An automorphism of an extraspecial p-group acting trivially on its Frattini quotient is inner Proposition
• Dih(C₄) and Q₈ are extraspecial of order 8, with six and two solutions of x²=1 respectively Proposition
• Every conjugacy class of an extraspecial p-group outside the centre has exactly p elements Proposition
• Order, centre and derived subgroup of a central product Proposition
• The Heisenberg group of order p³ is extraspecial, and for odd p it has exponent p Proposition
• The modular group of order p³ is extraspecial, of exponent p² when p is odd Proposition
• Three equivalent descriptions of an extraspecial p-group Proposition
• Φ(P× Q)=Φ(P)×Φ(Q) for finite p-groups Proposition
• A central product of extraspecial p-groups identified along their centres is extraspecial Theorem
• Abelianisation is left adjoint to the inclusion of abelian groups Theorem
• For each prime there are exactly two nonabelian groups of order p³ up to isomorphism Theorem
• G/N is abelian if and only if [G,G]⊆ N Theorem
• Iwasawa's simplicity criterion for primitive actions Theorem
• The abelianisation of a free group on X is a free abelian group on X Theorem
• The derived subgroup is characteristic and the abelianization is universal Theorem
• Φ(P)=P'Pᵖ for a finite p-group Theorem

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Encyclopedia of Mathematics, Commutator subgroup (standard reference, not scraped)