Group of finite composition length: Difference between revisions

Revision as of 14:58, 15 November 2008

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This is a variation of finiteness (groups)|Find other variations of finiteness (groups) |

This property makes sense for infinite groups. For finite groups, it is always true

Definition

A group is said to have finite composition length if it satisfies the following equivalent conditions:

It possesses a composition series of finite length, viz., a subnormal series such that all the successive quotients are simple groups.
Every subnormal series (without repeated terms) can be refined to a composition series of finite length.
It satisfies both the ascending chain condition on subnormal subgroups and the descending chain condition on subnormal subgroups.
There is an upper bound on the length of any subnormal series for the group (this upper bound equals the composition length, i.e., the length of any composition series).

Relation with other properties

Stronger properties

Finite group

Weaker properties

Metaproperties

Normal subgroups

This group property is normal subgroup-closed: any normal subgroup (and hence, any subnormal subgroup) of a group with the property, also has the property
View other normal subgroup-closed group properties

Any normal subgroup of a group of finite composition length again has finite composition length. In fact, if $N$ is a normal subgroup of a group $G$ , the composition length of $G$ equals the sum of the composition lengths of $N$ and $G/N$ .

@@ Line 10: / Line 10: @@
 # It possesses a [[defining ingredient::composition series]] of finite length, viz., a [[subnormal series]] such that all the successive quotients are [[defining ingredient::simple group]]s.
+# Every [[defining ingredient::subnormal series]] (without repeated terms) can be refined to a composition series of finite length.
 # It satisfies both the [[defining ingredient::group satisfying ascending chain condition on subnormal subgroups|ascending chain condition on subnormal subgroups]] and the [[defining ingredient::group satisfying descending chain condition on subnormal subgroups|descending chain condition on subnormal subgroups]].
+# There is an upper bound on the length of any [[defining ingredient::subnormal series]] for the group (this upper bound equals the [[composition length]], i.e., the length of any composition series).
 ==Relation with other properties==
@@ Line 24: / Line 26: @@
 * [[Stronger than::Group satisfying ascending chain condition on normal subgroups]]
 * [[Stronger than::Group satisfying descending chain condition on normal subgroups]]
+* [[Stronger than::Group of finite chief length]]
+* [[Stronger than::Group in which all subnormal subgroups have a common bound on subnormal depth]]
+==Metaproperties==
+{{normal subgroup-closed}}
+Any normal subgroup of a group of finite composition length again has finite composition length. In fact, if <math>N</math> is a normal subgroup of a group <math>G</math>, the composition length of <math>G</math> equals the sum of the composition lengths of <math>N</math> and <math>G/N</math>.