Central series

A central series is a chain of normal subgroups whose successive quotients lie in the centers of the corresponding quotient groups.

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- Central-series comparison theorem Central series

Central-series comparison theorem ( $Γ_{r - i + 1} (G) \leq G_{i} \leq Z_{i} (G)$ )

r

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

, induction from the top gives

Γ_{r - i + 1} (G) \leq G_{i}

, and induction from the bottom gives

G_{i} \leq Z_{i} (G)

. Consequently the lower central series terminates at

Γ_{c + 1} = 1

exactly when the upper central series reaches

Z_{c} = G

.

// Target: algebra.bigb

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In group theory, a branch of abstract algebra, a **central series** is a specific type of series of subgroups associated with a given group. It provides a way to study the structure of a group by breaking it down into simpler components.

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