Math 2.0

Group theory and generalizations Proof

Proof of the uniqueness of inverses

Builds on GroupInverse element

Proof

  1. Let aa be an element of a group GG with identity ee. Let bb and cc both be inverses of aa, so ba=eba = e and ac=eac = e.
  2. Then b=be=b(ac)b = be = b(ac).
  3. By associativity, b(ac)=(ba)cb(ac) = (ba)c.
  4. Since ba=eba = e, (ba)c=ec=c(ba)c = ec = c.
  5. Thus b=cb = c, and aa has only one inverse.

Idea

Multiply one inverse by the element and then by the other inverse. Associativity lets the product collapse in two ways.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptIdentity element, definition, no evidence levelIdentity elementdefinitionInverse element, definition, no evidence levelInverse elementdefinitionProof of the uniqueness of inverses, proof, no evidence levelProof of the uniquene…proof
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  1. Binary operation definition
  2. Group concept
  3. Identity element definition
  4. Inverse element definition
  5. Proof of the uniqueness of inverses proof

Where this leads

14 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Group_(mathematics), Uniqueness of inverses
About this page
TagRW1ZF04CRF
URIm2://RW1ZF04CRF
Kindproof
WritingW0 stub
MSC20A05
ProvesUniqueness of inverses
UsesGroup, Inverse element
MethodsDirect proof

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