Index of a subgroup
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This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Definition
Symbol-free definition
The index of a subgroup in a group is the following equivalent things:
- The number of left cosets of the subgroup
- The number of right cosets of the subgroup
The collection of left cosets is sometimes termed the coset space, so in this language, the index of a subgroup is the cardinality of its coset space.
Definition with symbols
Given a subgroup of a group , the index of in , denoted , is defined in the following ways:
- It is the number of left cosets of in , i.e. the number of sets of the form .
- It is the number of right cosets of in , i.e. the number of sets of the form .
The collection of left cosets of in is sometimes termed the coset space, and is denoted . With this notation, the index of in , is the cardinality .
Equivalence of definitions
The equivalence of definitions follows from the fact that there is a natural bijection between the collection of left cosets of a subgroup, and the collection of its right cosets, given by the map
Further information: Left and right coset spaces are naturally isomorphic
Further note for finite groups
When the group is finite, then by Lagrange's theorem, the index of a subgroup is the ratio of the order of the group to the order of the subgroup.
Facts
Multiplicativity of the index
Further information: Index is multiplicative If , then we have:
In other words, the number of cosets of in equals the number of cosets of in , times the number of cosets of in .
In fact, more is true. We can set up a bijection as follows:
However, this bijection is not a natural one, and, in order to define it, we first need to choose a system of coset representatives of .
Effect of intersection on the index
Further information: Conjugate-intersection index theorem If and are two subgroups of , then the index of is bounded above by the product of the indices of and of .
This follows as a consequence of the product formula. Note that equality holds if and only if .
Note that in case and are conjugate subgroups of index , the index of is bounded above by .
Related notions
For double cosets and multicosets
Related subgroup properties
References
Textbook references
- Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 57, Point (6.8) (definition in paragraph, defined as number of left cosets)
- Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 90 (formal definition, defined as number of left cosets)