# Subnormal series

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

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## Definition

### Definition for finite length

A subnormal series is a subgroup series where each member of the series is normal in the next one containing it. In symbols:

- A descending series:

of subgroups of a group is termed a subnormal series if is a normal subgroup of for .

- An ascending series:

of subgroups of a group is termed a subnormal series if each is a normal subgroup of .

Note that the subnormal series must have its largest member equal to the whole group. In some contexts, the term *subnormal series* refers to a subnormal series that terminates at the trivial subgroup. Note that any subnormal series of a group can be extended to such a subnormal series by adding the trivial group at the end.

### Definition for infinite length

`Further information: Subnormal series of infinite length`

## Relation with other properties

### Stronger properties

### Related subgroup properties

- A subnormal subgroup is a subgroup for which there is a subnormal series of finite length starting at the subgroup and ending at the whole group.
- An ascendant subgroup is a subgroup for which there is an ascending subnormal series of possibly infinite length starting at the subgroup and ending at the whole group.
- A descendant subgroup is a subgroup for which there is a descending subnormal series of possibly infinite length starting at the subgroup and ending at the whole group.
- A serial subgroup is a subgroup for which there is a subnormal series of possibly infinite length starting at the subgroup and ending at the whole group.