Subnormal series

In mathematics, specifically group theory, a subgroup series is a chain of subgroups:

Subgroup series can simplify the study of a group to the study of simpler subgroups and their relations, and several subgroup series can be invariantly defined and are important invariants of groups. A subgroup series is used in the subgroup method.

Subgroup series are a special example of the use of filtrations in abstract algebra.

A subnormal series (also normal series, normal tower, subinvariant series, or just series) of a group G is a sequence of subgroups, each a normal subgroup of the next one. In a standard notation

There is no requirement made that A_i be a normal subgroup of G, only a normal subgroup of A_i+1. The quotient groups A_i+1/A_i are called the factor groups of the series.

If in addition each A_i is normal in G, then the series is called a normal series, when this term is not used for the weaker sense, or an invariant series.

A series with the additional property that A_i ≠ A_i+1 for all i is called a series without repetition; equivalently, each A_i is a proper subgroup of A_i+1. The length of a series is the number of strict inclusions A_i < A_i+1. If the series has no repetition the length is n.

For a subnormal series, the length is the number of nontrivial factor groups. Every (nontrivial) group has a normal series of length 1, namely ${\displaystyle 1\triangleleft G}$, and any proper normal subgroup gives a normal series of length 2. For simple groups, the trivial series of length 1 is the longest subnormal series possible.

...
Wikipedia