a characterization of groups


idempotent element


A non-empty semigroupPlanetmathPlanetmath S is a group if and only if for every xS there is a unique yS such that xyx=x.


Suppose that S is a non-empty semigroup, and for every xS there is a unique yS such that xyx=x. For each xS, let x denote the unique element of S such that xxx=x. Note that x(xxx)x=(xxx)xx=xxx=x, so, by uniqueness, xxx=x, and therefore x′′=x.

For any xS, the element xx is idempotentMathworldPlanetmath (http://planetmath.org/Idempotency), because (xx)2=(xxx)x=xx. As S is nonempty, this means that S has at least one idempotent element. If iS is idempotent, then ix=ix(ix)ix=ix(ix)iix, and so (ix)i=(ix), and therefore (ix)=(ix)(ix)′′(ix)=(ix)ix(ix)=(ix)x(ix), which means that ix=(ix)′′=x. So every idempotent is a left identityPlanetmathPlanetmath, and, by a symmetricPlanetmathPlanetmath argument, a right identity. Therefore, S has at most one idempotent element. Combined with the previous result, this means that S has exactly one idempotent element, which we will denote by e. We have shown that e is an identityPlanetmathPlanetmathPlanetmath, and that xx=e for each xS, so S is a group.

Conversely, if S is a group then xyx=x clearly has a unique solution, namely y=x-1. ∎

Note. Note that inverse semigroups do not in general satisfy the hypothesisMathworldPlanetmath of this theorem: in an inverse semigroup there is for each x a unique y such that xyx=x and yxy=y, but this y need not be unique as a solution of xyx=x alone.

Title a characterization of groups
Canonical name ACharacterizationOfGroups
Date of creation 2013-03-22 14:45:08
Last modified on 2013-03-22 14:45:08
Owner yark (2760)
Last modified by yark (2760)
Numerical id 10
Author yark (2760)
Entry type Theorem
Classification msc 20A05
Related topic Group
Related topic RegularSemigroup
Related topic AlternativeDefinitionOfGroup