Subnormal series

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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

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.