A group may have more than one composition series.

A composition series under this action is exactly a chief series.

When just 21, Zassenhaus was studying composition series in group theory.

That is, there are no additional subgroups which can be "inserted" into a composition series.

In this generality, one obtains a composition series, rather than a direct sum.

A composition series may not even exist, and when it does, it need not be unique.

The maximum number of proper subobjects in any such composition series is called the length of A.

For example, has no composition series.

In group theory the Jordan-Hölder theorem on composition series is a basic result.

In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces.

Composition series may thus be used to define invariants of finite groups and Artinian modules.

A composition series is a maximal subnormal series.

Every finite group has a composition series, but not every infinite group has one.

If A has a composition series, the integer n only depends on A and is called the length of A.

Equivalently, a chief series is a composition series of the group G under the action of inner automorphisms.

In abelian groups, chief series and composition series are identical, as all subgroups are normal.

She is known most for her Composed Improvisation and Improvised Composition series.

The only composition series of is (where is the alternating group on five letters, also known as the icosahedral group).

If M is Artinian (or Noetherian), then has a finite composition series.

The Grothendieck group ignores the order in a composition series and views every finite length module as a formal sum of simple modules.

Compare composition series in Jordan-Hölder theorem.

Then there exists a composition series with infinite cyclic factors, which induces a bijection (not though necessarily a homomorphism).

The requirement that a group have a composition series is analogous to that of compactness in topology, and can sometimes be too strong a requirement.

Stacking the composition series from the end to end, we obtain a composition series for M.

In essence, each field extension L/K corresponds to a factor group in a composition series of the Galois group.