Editing Subgroup series (section)

===Ascending series, descending series===
Series can be notated in either ascending order:
:<math>1 = A_0\leq A_1\leq \cdots \leq A_n = G</math>
or descending order:
:<math>G = B_0\geq B_1\geq \cdots \geq B_n = 1.</math>

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
:<math>1 = A_0\leq A_1\leq \cdots \leq G</math>
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
:<math>G = B_0\geq B_1\geq \cdots \geq 1</math>
has a largest term, but no smallest proper term.

Further, given a recursive formula for producing a series, the terms produced are either ascending or descending, and one calls the resulting series an ascending or descending series, respectively. For instance the [[derived series]] and [[lower central series]] are descending series, while the [[upper central series]] is an ascending series.