Series (group theory): Difference between revisions
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In [[group theory]], a '''series''' is a [[chain (mathematics)]] of [[subgroup]]s of a [[group (mathematics)|group]] ordered by [[subset]] [[inclusion]]. The structure of the group is closely related to the existence of series with particular properties. | In [[group theory]], a '''series''' is a [[chain (mathematics)]] of [[subgroup]]s of a [[group (mathematics)|group]] ordered by [[subset]] [[inclusion]]. The structure of the group is closely related to the existence of series with particular properties. | ||
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:<math>G = A_0 \supseteq A_1 \supseteq \cdots \supseteq A_n . \, </math> | :<math>G = A_0 \supseteq A_1 \supseteq \cdots \supseteq A_n . \, </math> | ||
If the final group in the series is ''H'' we speak of a series '''from''' ''G'' '''to''' ''H''. | |||
The series is '''subinvariant''' or '''subnormal''' if each subgroup is a normal subgroup of its predecessor, <math>A_i \triangleleft A_{i-1}</math>. A subinvariant series in which each subgroup is a maximal normal subgroup of its predecessor is a '''composition series'''. | The series is '''subinvariant''' or '''subnormal''' if each subgroup is a normal subgroup of its predecessor, <math>A_i \triangleleft A_{i-1}</math>. A subinvariant series in which each subgroup is a maximal normal subgroup of its predecessor is a '''composition series'''. | ||
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==References== | ==References== | ||
* {{cite book | author=Marshall Hall jr | title=The theory of groups | publisher=Macmillan | location=New York | year=1959 | pages=123-124 }} | * {{cite book | author=Marshall Hall jr | title=The theory of groups | publisher=Macmillan | location=New York | year=1959 | pages=123-124 }}[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 06:01, 17 October 2024
In group theory, a series is a chain (mathematics) of subgroups of a group ordered by subset inclusion. The structure of the group is closely related to the existence of series with particular properties.
A series is a linearly ordered chain of subgroups of a given group G beginning with the group G itself:
If the final group in the series is H we speak of a series from G to H.
The series is subinvariant or subnormal if each subgroup is a normal subgroup of its predecessor, . A subinvariant series in which each subgroup is a maximal normal subgroup of its predecessor is a composition series.
The series is invariant or normal if each subgroup is a normal subgroup of the whole group. A subinvariant series in which each subgroup is a normal subgroup of G maximal subject to being a proper subgroup of its predecessor is a principal series or chief series.
References
- Marshall Hall jr (1959). The theory of groups. New York: Macmillan, 123-124.