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Let  be a group.
An ascending series of  is a family
 of subgroups of  , where  is an ordinal, such that  and  , and
 for all
 , and
whenever
 is a limit ordinal.
Note that this is a generalization of the concept of a subnormal series. Compare also the dual concept of a descending series.
Given an ascending series
 , the subgroups  are called the terms of the series and the quotients
 are called the factors of the series.
A subgroup of  that is a term of some ascending series of  is called an ascendant subgroup of  . The notation
 is sometimes used to indicate that  is an ascendant subgroup of  .
The groups in which every subgroup is ascendant are precisely the groups that satisfy the normalizer condition. Groups in which every cyclic subgroup is ascendant are called Gruenberg groups. It can be shown that in a Gruenberg group, every finitely generated subgroup is ascendant and nilpotent (and so, in particular, Gruenberg groups are locally nilpotent).
An ascending series of  in which all terms are normal in  is called an ascending normal series.
Let
 be a property of groups. A group is said to be hyper-
 if it has an ascending normal series whose factors all have property
 . So, for example, a hyperabelian group is a group that has an ascending normal series with abelian factors. Hyperabelian groups are sometimes called  -groups.
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