Subnormal series

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.

Stronger properties

 * Weaker than::Normal series
 * Weaker than::Characteristic series
 * Weaker than::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.