Proof of the uniqueness of inverses

Proof of the uniqueness of inverses
Tag	`RW1ZF04CRF`
URI	`m2://RW1ZF04CRF`
Kind	proof
Writing	W0 stub
MSC	20A05
Proves	Uniqueness of inverses
Uses	Group, Inverse element
Methods	Direct proof

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Proof

Let $a$ be an element of a group $G$ with identity $e$ . Let $b$ and $c$ both be inverses of $a$ , so $ba = e$ and $ac = e$ .
Then $b = be = b(ac)$ .
By associativity, $b(ac) = (ba)c$ .
Since $ba = e$ , $(ba)c = ec = c$ .
Thus $b = c$ , and $a$ 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.

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Binary operation definition
Group concept
Identity element definition
Inverse element definition
Proof of the uniqueness of inverses proof

What depends on this

12 nodes use this node, directly or through other nodes.

Inner automorphism group is isomorphic to G modulo its centre 05HYGWFG63
Homomorphisms preserve identity and inverses 08FKQD7NJ3
The kernel is a normal subgroup 0C9ZKYSPYR
First isomorphism theorem 0CXXJNG5B6
Uniqueness of inverses 0D9NDGE9KP
A homomorphism is injective iff its kernel is trivial 0N2P0XRTDF
Second isomorphism theorem 0PPP2J2DJV
Order of the alternating group 0SZ1KMNWVE
Third isomorphism theorem 0TJ1V9DDPC
Proof that the kernel is a normal subgroup AY60KQR8RT
Proof that a homomorphism is injective iff its kernel is trivial D05NAAC3XK
Proof of the first isomorphism theorem H4QSPK0XTZ

Sources

Web https://en.wikipedia.org/wiki/Group_(mathematics), Uniqueness of inverses

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