Group theory and generalizations Proposition
Cosets have the same cardinality as the subgroup
Builds on Left cosetRight coset
Statement
Let be a group, let be a subgroup of , and let . Then the map is a bijection from to the left coset . The map is a bijection from to the right coset . In particular, .
Remarks
The inverse of is . This fact and the partition of into left cosets give the proof of Lagrange's theorem.
Reading path
To understand this page from the ground up, read these 5 pages in order. The foundations come first.
Show the path as a graph
- no evidence level
- Group concept
- Subgroup definition
- Left coset definition
- Right coset definition
- Cosets have the same cardinality as the subgroup statement
Where this leads
31 pages build on this one.
- A finite group is nilpotent iff it is the product of its Sylow subgroups
- Order of a group is the index times the order of the subgroup
- Groups of prime order are cyclic
- Product formula for the order of HK
- Order of an element divides the order of the group
- Fixed points of a p-group action
- Second Sylow theorem
- Groups of order p squared are abelian
- The number of Sylow subgroups is the index of the normaliser
- The alternating group A5 is simple
- Abelian simple groups have prime order
- Cauchy's theorem
- The number of Sylow subgroups divides the index
- Converse of Lagrange's theorem fails for the alternating group A4
- Third Sylow theorem
- Frattini argument
- Size of an orbit times the size of the stabiliser
- First Sylow theorem
- Groups of order pq
- Burnside's lemma
- g to the order of the group is the identity
- Lagrange's theorem
- Galois group of a prime degree polynomial with two nonreal roots
- Nichols–Zoeller theorem
- Hall's theorem on solvable groups
- Proof that the order of an element divides the order of the group
- Hall subgroup
- Sylow theorems
- Frattini subgroup
- Proof that groups of prime order are cyclic
- Proof of Lagrange's theorem
Further reading
- Web https://en.wikipedia.org/wiki/Coset, Properties
About this page
| Tag | 0DKMBWTD7P |
|---|---|
| URI | m2://0DKMBWTD7P |
| Kind | statement, proposition |
| Writing | W0 stub |
| MSC | 20A05 |
| Status | This copy of the site has no build data. Run m2 status 0DKMBWTD7P. |
| Uses | Left coset, Right coset |
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