The lower central series descends from the whole group toward the identity, retaining successively longer commutators. It is the canonical series for nilpotency class and commutator weight. [@isaacs2008]
Definition
\(\gamma_2(G)=G'\). A left-normed commutator \([x_1,\ldots,x_i]\) lies in \(\gamma_i(G)\).
Core result
The minimality statement follows inductively from the definition. If \(\gamma_{c+1}=1\), reading the factors backward shows that each \(\gamma_i/\gamma_{i+1}\) is central in the relevant quotient. Comparison with the upper central series gives nilpotence. See Sections 4C-4D.
Structural properties
- Every \(\gamma_i(G)\) is fully invariant and is preserved by all endomorphisms.
- \([\gamma_i(G),\gamma_j(G)]\leq\gamma_{i+j}(G)\).
- Quotients satisfy \(\gamma_i(G/N)=\gamma_i(G)N/N\).
- The derived series satisfies \(G^{(i)}\leq\gamma_{2^i}(G)\), so nilpotent groups are solvable.
- Residual nilpotence means \(\bigcap_i\gamma_i(G)=1\) and is weaker than finite-step nilpotence.
The lower central series assigns commutator weights and makes Hall–Witt identities and collection processes computable layer by layer.
Example and boundary
Knowledge network
The upper central series approaches from the other end, the nilpotent-group core article proves equivalence, and the derived series supplies the solvability comparison.
Proof and sources
Definitions, commutator weights, and the nilpotence criterion are in Sections 4C-4D; longer identities are supported by Hall–Witt and the three-subgroups lemma. [@isaacs2008]
