Subgroup series

In mathematics, specifically group theory, a subgroup series of a group is a chain of subgroups: where is the trivial subgroup. 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 Ai be a normal subgroup of G, only a normal subgroup of Ai +1. The quotient groups Ai +1/Ai are called the factor groups of the series. If in addition each Ai 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 Ai ≠ Ai +1 for all i is called a series without repetition; equivalently, each Ai is a proper subgroup of Ai +1. The length of a series is the number of strict inclusions Ai < Ai +1. If the series has no repetition then the length is n. For a subnormal series, the length is the number of non-trivial factor groups. Every nontrivial group has a normal series of length 1, namely , and any nontrivial 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. Series can be notated in either ascending order: or descending order: For a given finite series, there is no distinction between an "ascending series" or "descending series" beyond notation. For infinite series however, there is a distinction: the ascending series has a smallest term, a second smallest term, and so forth, but no largest proper term, no second largest term, and so forth, while conversely the descending series has a largest term, but no smallest proper term.