# Central series

From Groupprops

This article defines a property that can be evaluated for a subgroup seriesView a complete list of properties of subgroup series

## Definition

### Central series of finite length

This is the default meaning of the term *central series*.

is termed a **central series** if it satisfies the following conditions:

- It is a normal series: every is normal in
- For every , is contained in the center of .

Equivalently, it should satisfy the condition that for every :

### Descending central series of possibly infinite or transfinite length

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### Ascending central series of possibly infinite or transfinite length

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### Equivalence of definitions

`Further information: Equivalence of definitions of central series`

## Relation with other properties

### Stronger properties

### Weaker properties

- Normal series:
`For full proof, refer: Central series implies normal series` - Subnormal series