Ascending Central Series and Nilpotent Groups

Ascending Central Series of G

Let G be a group. The center C(G) of G is a normal subgroup. Let C_2(G) be the inverse image of C(G/C(G)) under the canonical projection G\to G/C(G). By Correspondence Theorem, C_2(G) is normal in G and contains C(G).

Continue this process by defining inductively: C_1(G)=C(G) and C_i(G) is the inverse image of C(G/C_{i-1}(G)) under the canonical projection G\to G/C_{i-1}(G).

Thus we obtain a sequence of normal subgroups of G, called the ascending central series of G: \displaystyle \langle e\rangle<C_1(G)<C_2(G)<\cdots.

Nilpotent Group

A group G is nilpotent if C_n(G)=G for some n.

Abelian Group is Nilpotent

Every abelian group G is nilpotent since G=C(G)=C_1(G).

Every finite p-group is nilpotent (Proof)

G and all its nontrivial quotients are p-groups, and therefore have non-trivial centers.

Hence if G\neq C_i(G), then G/C_i(G) is a p-group, and C(G/C_i(G)) is non-trivial. Thus C_{i+1}(G), the inverse image of C(G/C_i(G)) under \pi:G\to G/C_i(G), strictly contains C_i(G).

Since G is finite, C_n(G) must be G for some n.


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