Group theory and generalizations Definition
Subgroup
Builds on Group
Definition
Let be a group with identity element . A subgroup of is a subset with these three properties. The identity is in . For all , the product is in . For each , the inverse is in . The notation means that is a subgroup of .
Remarks
A subgroup is a group with the operation of restricted to , and its identity is the identity of . Each group has the subgroups and . The even integers form a subgroup of . The natural numbers do not form a subgroup of , because is not a natural number.
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Where this leads
174 pages build on this one.
- A finite group is nilpotent iff it is the product of its Sylow subgroups
- Image of a homomorphism
- Semidirect product
- Characteristic subgroup
- Normality via left and right cosets
- Extensions of solvable groups are solvable
- Quotients of solvable groups are solvable
- Infinite cyclic group
- Order of a cyclic subgroup equals the order of its generator
- Subgroups of a finite cyclic group correspond to divisors of its order
- Order of a group is the index times the order of the subgroup
- Index of a subgroup
- Cyclic groups are abelian
- The dihedral group of order 8
- Stabilisers of points in the same orbit are conjugate
- Left coset
- Inner automorphism group is isomorphic to G modulo its centre
- Groups of prime order are cyclic
- Subgroup criterion
- Subgroups of solvable groups are solvable
- Right coset
- Classification of finite simple groups
- Cyclic groups of the same order are isomorphic
- Classification of finite abelian groups
- Finite p-groups are nilpotent
- Class equation
- Product of cyclic groups is cyclic iff the orders are coprime
- Fixed points of an action
- Lower central series
- Product formula for the order of HK
- Commutator subgroup
- Order of an element divides the order of the group
- Subnormal series
- Fixed points of a p-group action
- Second Sylow theorem
- Dihedral group
- The symmetric group S3
- 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
- The symmetric group is generated by transpositions
- Simplicity of the alternating group
- Normaliser
- Action on left cosets
- Cyclic group
- Lattice of subgroups
- Order of the dihedral group
- Simple group
- The kernel is a normal subgroup
- First isomorphism theorem
- Conjugate subgroup
- 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
- Cyclic group of order n
- Automorphism group of a cyclic group
- Universal property of the quotient group
- The alternating group A5 is simple
- Nilpotent groups are solvable
- Abelianisation
- Finitely generated group
- The symmetric group on five or more letters is not solvable
- Abelian simple groups have prime order
- Cauchy's theorem
- Subgroups of abelian groups are normal
- Sylow p-subgroup
- Subgroups of cyclic groups are cyclic
- The alternating group is generated by three-cycles
- The number of Sylow subgroups divides the index
- Generators of a group
- A finite abelian group is cyclic iff its exponent equals its order
- A quotient is abelian iff the subgroup contains the commutator subgroup
- Composition series
- Product of two subgroups
- Solvable group
- Action by conjugation
- Converse of Lagrange's theorem fails for the alternating group A4
- Second isomorphism theorem
- Centraliser
- Internal direct product
- Third Sylow theorem
- Von Dyck's theorem
- The group of units of Z/pZ is cyclic
- Frattini argument
- Presentation of a group
- Abelian groups are solvable
- Derived series
- Composition factors
- Subgroup generated by a subset
- Stabiliser
- 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
- Subgroups of the integers
- Characteristic subgroups are normal
- Order of the alternating group
- Normal core
- Third isomorphism theorem
- Product of a subgroup and a normal subgroup is a subgroup
- Subgroups of index two are normal
- First Sylow theorem
- Size of a conjugacy class is the index of the centraliser
- Normal subgroup
- Number of elements of each order in a cyclic group
- Groups of order pq
- Cayley graph
- Integers modulo n as a quotient group
- Free abelian group
- Cyclic subgroup generated by an element
- Quotient group
- Burnside's lemma
- g to the order of the group is the identity
- Tower law for indices
- Intersection of subgroups is a subgroup
- 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
- Space of connections modulo gauge
- Proof that cyclic groups are abelian
- The even integers form a subgroup of the integers
- Hall's theorem on solvable groups
- Krull–Schmidt theorem
- Rational equivalence
- Proof that the kernel is a normal subgroup
- Ebin's slice theorem
- 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
- Integers modulo n form a cyclic group
- Sylow theorems
- Frattini subgroup
- Lattice-ordered group
- Proof that groups of prime order are cyclic
- Proof of the first isomorphism theorem
- Jordan–Hölder theorem for modules
- Galois group of a finite field extension
- The cosets of a subgroup that is not normal do not form a group
- Algebraic cycle
- Algebraic K-theory of schemes
- Cyclic extension
- 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
- The odd integers are not a subgroup of the integers
- Galois' theorem on solvability
- Moduli space of instantons
- Torsion-free group
- Slice theorem
- Proof of Lagrange's theorem
- Finite lattice representation problem
- 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/Subgroup, lead section
- Web https://en.wikipedia.org/wiki/Subgroup, Basic properties of subgroups
About this page
| Tag | 0N0PXCTYA7 |
|---|---|
| URI | m2://0N0PXCTYA7 |
| Kind | definition |
| Writing | W2 usable |
| MSC | 20A05 |
| Defines | Subgroup |
| Uses | Group |
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Used by
- Image of a homomorphism
- Characteristic subgroup
- Left coset
- Subgroup criterion
- Subgroups of solvable groups are solvable
- Right coset
- Normaliser
- Lattice of subgroups
- Conjugate subgroup
- Sylow p-subgroup
- Subgroups of cyclic groups are cyclic
- Product of two subgroups
- Centraliser
- Subgroup generated by a subset
- Stabiliser
- Torsion subgroup
- Normal subgroup
- Cyclic subgroup generated by an element
- Intersection of subgroups is a subgroup
- The even integers form a subgroup of the integers
- The odd integers are not a subgroup of the integers