Difference between revisions of "Group with operators"
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In practice, we deal with general groups with operators infrequently. | In practice, we deal with general groups with operators infrequently. | ||
| − | However, many structures& | + | However, many structures—groups and [[module]]s (including [[ring]]s, |
| − | [[field]]s, and [[vector space]]s)& | + | [[field]]s, and [[vector space]]s)—are special cases of this |
| − | general structure, and we can prove many results& | + | general structure, and we can prove many results—for example, |
| − | the [[Jordan-Hölder Theorem]]& | + | the [[Jordan-Hölder Theorem]]—about groups with operators in |
general; we then avoid repeated proofs of the same results in different | general; we then avoid repeated proofs of the same results in different | ||
fields. | fields. | ||
Latest revision as of 23:30, 18 May 2009
A group with operators is a group 
with a set of operators
such that each 
is associated with a group
endomorphism 
on .
By abuse of notation, we usually refer
to as simply 
, and we write 
as 
,
when is written multiplicatively; when is written additively,
we usually write 
, or simply 
.
A subgroup of a group with operators is called a stable subgroup if it is closed under the action of the operators. It is called a normal stable subgroup if it is a normal subgroup and a stable subgroup.
In practice, we deal with general groups with operators infrequently. However, many structures—groups and modules (including rings, fields, and vector spaces)—are special cases of this general structure, and we can prove many results—for example, the Jordan-Hölder Theorem—about groups with operators in general; we then avoid repeated proofs of the same results in different fields.
Emmy Noether was responsible for much of the study of groups with operators.
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