Subgroup commutators and the lower central series

Depends on

• Commutators $[g,h]=ghg^{-1}h^{-1}$ and the commutator subgroup $[G,G]$
• The subgroup $\langle S \rangle$ generated by a subset, the cyclic subgroup $\langle g \rangle$, and cyclic groups

Used by

• An extraspecial p-group is nilpotent of class exactly two and its derived subgroup has order p Corollary
• An extraspecial p-group is the product of two maximal abelian subgroups meeting in its centre Corollary
• Frobenius automizer criterion for p nilpotence Corollary
• Nilpotent groups, and in particular finite p-groups, are solvable Corollary
• Internal central products of a finite family of subgroups Definition
• The homogeneous dimension of a finitely generated nilpotent group Definition
• The dihedral group of order eight is nilpotent of class two Example
• The integral Heisenberg group is nilpotent of class two Example
• Central factors are equivalent to adjacent commutator containments Lemma
• Commutator identities in a group whose derived subgroup is central Lemma
• Commutator product identities in the fixed convention Lemma
• Distinct normal Sylow subgroups centralize one another Lemma
• Fusion control forces trivial Sylow intersection with the p residual Lemma
• Normal p subgroup has proper commutator in a p group Lemma
• A central extension of a class-c nilpotent group is nilpotent of class at most c+1 Theorem
• Internal central products are the images of external ones Theorem
• Nilpotence via central series, the upper central series, and the lower central series Theorem
• Subgroups, quotients, and finite direct products of nilpotent groups are nilpotent Theorem

Dependency tree · two levels

6 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

• J. S. Milne, Group Theory, Chapter 6 (standard reference, not scraped)
• K. Conrad, Subgroup Series I (standard reference, not scraped)
• K. Igusa, Notes on Jordan-Hölder, section 5 (standard reference, not scraped)