Math 2.0

Group theory and generalizations Proposition

Homomorphisms preserve identity and inverses

Builds on Group homomorphismCancellation lawsUniqueness of inverses

Statement

Let GG and HH be groups with identity elements eGe_G and eHe_H, and let φ ⁣:G→H\varphi \colon G \to H be a group homomorphism. Then φ(eG)=eH\varphi(e_G) = e_H, and φ(a−1)=φ(a)−1\varphi(a^{-1}) = \varphi(a)^{-1} for each a∈Ga \in G.

Remarks

The first equation follows from φ(eG)φ(eG)=φ(eG)=φ(eG)eH\varphi(e_G) \varphi(e_G) = \varphi(e_G) = \varphi(e_G) e_H and cancellation. The second follows from φ(a)φ(a−1)=φ(eG)=eH\varphi(a) \varphi(a^{-1}) = \varphi(e_G) = e_H and the uniqueness of inverses. A consequence is that φ(an)=φ(a)n\varphi(a^n) = \varphi(a)^n for each integer nn.

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
Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptIdentity element, definition, no evidence levelIdentity elementdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionInverse element, definition, no evidence levelInverse elementdefinitionProof of the cancellation laws, proof, no evidence levelProof of the cancella…proofProof of the uniqueness of inverses, proof, no evidence levelProof of the uniquene…proofCancellation laws, statement, no evidence levelCancellation lawsstatementUniqueness of inverses, statement, no evidence levelUniqueness of inversesstatementHomomorphisms preserve identity and inverses, statement, no evidence levelHomomorphisms preserv…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Binary operation definition
  2. Group concept
  3. Identity element definition
  4. Group homomorphism definition
  5. Inverse element definition
  6. Proof of the cancellation laws proof
  7. Proof of the uniqueness of inverses proof
  8. Cancellation laws statement
  9. Uniqueness of inverses statement
  10. Homomorphisms preserve identity and inverses statement

Where this leads

12 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Group_homomorphism, Properties
About this page
Tag08FKQD7NJ3
URIm2://08FKQD7NJ3
Kindstatement, proposition
WritingW0 stub
MSC20A05
StatusThis copy of the site has no build data. Run m2 status 08FKQD7NJ3.
UsesGroup homomorphism, Cancellation laws, Uniqueness of inverses

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