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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-7/1/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 7 / 1 / i / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

A central series is a finite chain of normal subgroups

1 = G_{0} \leq G_{1} \leq \dots \leq G_{r} = G

with

G_{i} / G_{i - 1} \leq Z (G / G_{i - 1})

for every

i \geq 1

. Equivalently, writing

[x, y] = x^{- 1} y^{- 1} x y

and

[A, B]

for the subgroup generated by these group commutators, the condition is

[G_{i}, G] \leq G_{i - 1}

. Centrality is relative to the whole quotient group, not just to the next subgroup in the chain.

A group is nilpotent if it admits a finite central series. The trivial group admits a series of length zero. For a nontrivial nilpotent group, the minimum possible length is its nilpotency class.

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