Corollary 1

Tag 05DK, permanent. Cite it as [Bourbaki, Tag 05DK].

Algebra, chapter I, § 6. Extensions, solvable groups, nilpotent groups

Let G be a nilpotent group, Z the centre of G and N a normal subgroup of G. If N ≠ {e}, then N ∩ Z ≠ {e}.

Let k be the largest index such that N^k ≠ {e}. The group N^k is contained in N. On the other hand, (G, N^k) ⊂ N^{k+1} = {e}; hence N^k is contained in the centre Z of G.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Corollary 2, § 6

Cites

Nothing.