Group theory and generalizations Definition

Lower central series

Builds on Commutator of two elements Subgroup generated by a subset Commutator subgroup

Definition

Let $G$ be a group. For subgroups $H$ and $K$ of $G$ , let $[H, K]$ be the subgroup generated by the commutators $[h, k]$ with $h \in H$ and $k \in K$ . The lower central series of $G$ is the sequence $\gamma_1(G) = G$ and $\gamma_{i+1}(G) = [\gamma_i(G), G]$ for $i \geq 1$ .

Remarks

The second term $\gamma_2(G) = [G, G]$ is the commutator subgroup. Each $\gamma_i(G)$ is normal in $G$ , and $\gamma_i(G)/\gamma_{i+1}(G)$ lies in the centre of $G/\gamma_{i+1}(G)$ . The terms of the derived series satisfy $G^{(i)} \subseteq \gamma_{i+1}(G)$ . For $S_3$ , the series is $S_3 \supseteq A_3 \supseteq A_3 \supseteq \cdots$ , so it never reaches $\{e\}$ .

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Group concept
Commutator of two elements definition
Subgroup definition
Subgroup criterion statement
Intersection of subgroups is a subgroup statement
Subgroup generated by a subset definition
Commutator subgroup definition
Lower central series definition

Tag	`08PA437800`
URI	`m2://08PA437800`
Aliases	`descending-central-series`
Kind	definition
Writing	W0 stub
MSC	20F14
Defines	Lower central series
Uses	Commutator of two elements, Subgroup generated by a subset, Commutator subgroup

Lower central series

Definition

Remarks

Reading path

Further reading

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