Difference between revisions of "Operator inverse"
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If the operation <math>G</math> is [[associative]] and an element has both a right and left inverse, these two inverses are equal. | If the operation <math>G</math> is [[associative]] and an element has both a right and left inverse, these two inverses are equal. | ||
===Proof=== | ===Proof=== | ||
− | Let <math>g</math> be the element with left inverse <math>h</math> and right inverse <math>h'</math>, so <math>G(h, g) = G(g, h') = e</math>. Then <math>G(G(h, g), h') = G(e, h') = h'</math>, by the properties of <math>e</math>. But by associativity, <math> | + | Let <math>g</math> be the element with left inverse <math>h</math> and right inverse <math>h'</math>, so <math>G(h, g) = G(g, h') = e</math>. Then <math>G(G(h, g), h') = G(e, h') = h'</math>, by the properties of <math>e</math>. But by associativity, <math>G(G(h, g), h') = G(h, G(g, h')) = G(h, e) = h</math>, so we do indeed have <math>h = h'</math>. |
===Corollary=== | ===Corollary=== | ||
If the operation <math>G</math> is associative, inverses are unique. | If the operation <math>G</math> is associative, inverses are unique. | ||
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+ | [[Category:Abstract algebra]] | ||
+ | [[Category:Definition]] |
Latest revision as of 10:40, 23 November 2007
Suppose we have a binary operation on a set , , and suppose this operation has an identity , so that for every we have . An inverse to under this operation is an element such that .
Thus, informally, operating by is the "opposite" of operating by -inverse.
If our operation is not commutative, we can talk separately about left inverses and right inverses. A left inverse of would be some such that , while a right inverse would be some such that .
Uniqueness (under appropriate conditions)
If the operation is associative and an element has both a right and left inverse, these two inverses are equal.
Proof
Let be the element with left inverse and right inverse , so . Then , by the properties of . But by associativity, , so we do indeed have .
Corollary
If the operation is associative, inverses are unique.