Subnormal series

This article defines a property that can be evaluated for a subgroup series

View a complete list of properties of subgroup series

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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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:

$G = H_0 \ge H_1 \ge H_2 \ge \dots \ge H_r$

of subgroups of a group $G$ is termed a subnormal series if $H_{i+1}$ is a normal subgroup of $H_i$ for $0 \le i \le r - 1$.

• An ascending series:

$H_0 \le H_1 \le H_2 \le \dots H_r = G$

of subgroups of a group $G$ is termed a subnormal series if each $H_i$ is a normal subgroup of $H_{i+1}$.

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

• Normal series
• Characteristic series
• Composition series

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.