Group theory and generalizations Proof
Proof that homomorphisms preserve identity and inverses
Builds on Group homomorphismCancellation lawsUniqueness of inverses
Proof
- Let be a group homomorphism, and let and be the identity elements of and .
- Since and is a homomorphism, .
- Since is the identity of , .
- By steps 2 and 3, . By the cancellation laws, .
- Let . Since , step 4 gives .
- By step 5 and the uniqueness of inverses, .
- Steps 4 and 6 are the two claims of the statement.
Idea
The identity is the only element of a group with , and has this property. Once the identity is preserved, is a right inverse of , and inverses in a group are unique.
Reading path
To understand this page from the ground up, read these 10 pages in order. The foundations come first. The path takes the proofs Proof of the uniqueness of inverses, Proof of the cancellation laws.
Show the path as a graph
- no evidence level
- Binary operation definition
- Group concept
- Identity element definition
- Group homomorphism definition
- Inverse element definition
- Proof of the cancellation laws proof
- Proof of the uniqueness of inverses proof
- Cancellation laws statement
- Uniqueness of inverses statement
- Proof that homomorphisms preserve identity and inverses proof
Where this leads
51 pages build on this one.
- A finite group is nilpotent iff it is the product of its Sylow subgroups
- Extensions of solvable groups are solvable
- Inner automorphism group is isomorphic to G modulo its centre
- Subgroups of solvable groups are solvable
- Homomorphisms preserve identity and inverses
- Groups of order p squared are abelian
- If G modulo its centre is cyclic then G is abelian
- Simplicity of the alternating group
- The kernel is a normal subgroup
- First isomorphism theorem
- Jordan–Hölder theorem
- Universal property of the abelianisation
- A subgroup whose index is the smallest prime divisor of the order is normal
- Automorphism group of a cyclic group
- The alternating group A5 is simple
- The symmetric group on five or more letters is not solvable
- The alternating group is generated by three-cycles
- A homomorphism is injective iff its kernel is trivial
- Second isomorphism theorem
- Von Dyck's theorem
- Order of the alternating group
- Fundamental theorem of Galois theory
- Third isomorphism theorem
- Gauss–Wantzel theorem
- Abel–Ruffini theorem
- Zassenhaus lemma
- Proof of the second isomorphism theorem
- Proof of the Jordan–Hölder theorem by induction on the length
- Proof that G is abelian if G modulo its centre is cyclic
- Proof of the description of the automorphism group of a cyclic group
- Proof that the inner automorphism group is isomorphic to G modulo its centre
- Proof of the third isomorphism theorem
- Proof of the order of the alternating group
- Proof that the kernel is a normal subgroup
- Proof that subgroups of solvable groups are solvable
- Proof that a subgroup whose index is the smallest prime divisor of the order is normal
- Proof that the alternating group is generated by three-cycles
- Proof that a homomorphism is injective iff its kernel is trivial
- Proof of von Dyck's theorem
- Proof of the universal property of the abelianisation
- Proof that groups of order p squared are abelian
- Proof of the first isomorphism theorem
- Proof of the simplicity of the alternating group by three-cycles
- Proof that extensions of solvable groups are solvable
- Galois group of a finite field extension
- Proof that a finite group is nilpotent iff it is the product of its Sylow subgroups
- Schreier refinement theorem
- Quintic equation
- Proof of the fundamental theorem of Galois theory
- Proof of the Gauss–Wantzel theorem
- Proof that the symmetric group on five or more letters is not solvable
Further reading
About this page
| Tag | W6V1XYH0K6 |
|---|---|
| URI | m2://W6V1XYH0K6 |
| Kind | proof |
| Writing | W0 stub |
| MSC | 20A05 |
| Proves | Homomorphisms preserve identity and inverses |
| Uses | Group homomorphism, Cancellation laws, Uniqueness of inverses |
| Changed | 4 November 2026 |
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