Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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Subgroup commutators and the lower central series
Definition
For subgroups , their subgroup commutator is where (Commutators and the commutator subgroup , The subgroup generated by a subset, the cyclic subgroup , and cyclic groups).
The lower central series is Each is characteristic in , and the series descends because whenever .
Depends on
Used by
- Nilpotent groups, and in particular finite p-groups, are solvable Corollary
- 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
- A central extension of a class-c nilpotent group is nilpotent of class at most c+1 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 · next 3 levels
Direct dependencies and their dependencies through the next three levels: 12 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)