Tour:Index of a subgroup
This article adapts material from the main article: index of a subgroup
This page is part of the Groupprops Guided tour for beginners (Jump to beginning of tour)
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WHAT YOU NEED TO DO:
- Read the two equivalent definitions of index (as number of left cosets, and number of right cosets)
- Convince yourself of why the two definitions are equivalent.
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
WHAT'S MORE: Some further facts about index of a subgroup, some of which you'll see soon in the tour, and others, that'll be seen much later. Ignore those that use unfamiliar terminology.
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.
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 .
This page is part of the Groupprops Guided tour for beginners (Jump to beginning of tour). If you found anything difficult or unclear, make a note of it; it is likely to be resolved by the end of the tour.
PREVIOUS: Left and right coset spaces are naturally isomorphic | UP: Introduction three (beginners) | NEXT: Lagrange's theorem