Group theory and generalizations Definition
Left coset
Builds on Subgroup
Definition
Let be a group, let be a subgroup of , and let . The left coset of with representative is the set .
Remarks
Two left cosets and are equal when , and they are disjoint otherwise. Thus the left cosets of form a partition of , and is one of them. In , the left cosets of are , and .
Reading path
To understand this page from the ground up, read these 3 pages in order. The foundations come first.
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- Group concept
- Subgroup definition
- Left coset definition
Where this leads
105 pages build on this one.
- A finite group is nilpotent iff it is the product of its Sylow subgroups
- Normality via left and right cosets
- Extensions of solvable groups are solvable
- Quotients of solvable groups are solvable
- Order of a group is the index times the order of the subgroup
- Index of a subgroup
- Inner automorphism group is isomorphic to G modulo its centre
- Groups of prime order are cyclic
- Subgroups of solvable groups are solvable
- Finite p-groups are nilpotent
- Class equation
- Product formula for the order of HK
- Order of an element divides the order of the group
- Subnormal series
- Fixed points of a p-group action
- Second Sylow theorem
- Outer automorphism group
- Groups of order p squared are abelian
- The number of Sylow subgroups is the index of the normaliser
- If G modulo its centre is cyclic then G is abelian
- Action on left cosets
- First isomorphism theorem
- Upper central series
- Cosets have the same cardinality as the subgroup
- Nilpotent group
- Jordan–Hölder theorem
- Orbit-stabiliser theorem
- Universal property of the abelianisation
- A subgroup whose index is the smallest prime divisor of the order is normal
- Burnside's p^a q^b theorem
- Universal property of the quotient group
- The alternating group A5 is simple
- Nilpotent groups are solvable
- Abelianisation
- The symmetric group on five or more letters is not solvable
- Abelian simple groups have prime order
- Cauchy's theorem
- The number of Sylow subgroups divides the index
- A quotient is abelian iff the subgroup contains the commutator subgroup
- Composition series
- Solvable group
- Converse of Lagrange's theorem fails for the alternating group A4
- Second isomorphism theorem
- Third Sylow theorem
- Von Dyck's theorem
- Frattini argument
- Presentation of a group
- Abelian groups are solvable
- Composition factors
- Torsion subgroup
- Structure theorem for finitely generated abelian groups
- A nontrivial finite p-group has nontrivial centre
- Size of an orbit times the size of the stabiliser
- Correspondence theorem
- Order of the alternating group
- Third isomorphism theorem
- Subgroups of index two are normal
- First Sylow theorem
- Size of a conjugacy class is the index of the centraliser
- Groups of order pq
- Integers modulo n as a quotient group
- Free abelian group
- Quotient group
- Burnside's lemma
- g to the order of the group is the identity
- Tower law for indices
- Abel–Ruffini theorem
- Zassenhaus lemma
- Lagrange's theorem
- Galois group of a prime degree polynomial with two nonreal roots
- Proof that subgroups of index two are normal
- The integers modulo the even integers
- The left cosets of a subgroup of order two in S3
- Nichols–Zoeller theorem
- Hall's theorem on solvable groups
- Krull–Schmidt theorem
- Rational equivalence
- Intersection multiplicity
- The cosets of 3Z in the integers
- Proof that the order of an element divides the order of the group
- Hall subgroup
- Fitting's lemma
- Sylow theorems
- Frattini subgroup
- Lattice-ordered group
- Proof that groups of prime order are cyclic
- Jordan–Hölder theorem for modules
- The cosets of a subgroup that is not normal do not form a group
- Algebraic cycle
- Algebraic K-theory of schemes
- Structure theorem for compactly generated LCA groups
- Left-orderable group
- Schreier refinement theorem
- Solvability of x^n - a
- Quintic equation
- Shafarevich's theorem on solvable Galois groups
- The alternating group A3 is normal in S3
- Length of a module
- Galois' theorem on solvability
- Torsion-free group
- Proof of Lagrange's theorem
- A subgroup of order two in S3 is not normal
- Divisor class group
- Solvable by radicals
- Orderable group
Further reading
- Web https://en.wikipedia.org/wiki/Coset, Definition
- Web https://en.wikipedia.org/wiki/Coset, Properties
About this page
| Tag | 05FD317V2W |
|---|---|
| URI | m2://05FD317V2W |
| Kind | definition |
| Writing | W0 stub |
| MSC | 20A05 |
| Defines | Left coset |
| Uses | Subgroup |
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What links here
Used by
- Normality via left and right cosets
- Index of a subgroup
- Action on left cosets
- Cosets have the same cardinality as the subgroup
- Orbit-stabiliser theorem
- Quotient group
- The left cosets of a subgroup of order two in S3
- The cosets of 3Z in the integers
- The cosets of a subgroup that is not normal do not form a group
- A subgroup of order two in S3 is not normal