+Signed integers 
*Signed integers 
+Group of geometric transformations of the plane. 
*Group of geometric transformations of the plane. 
One can build more generally the group of all distancepreserving transformations of a space or set of points. In this case, the proof of the group axioms is bit more complicated and interesting. (OpRgAg). 
One can build more generally the group of all distancepreserving transformations of a space or of a set of points in a space. In this case, the proof of the group axioms is a bit more interesting. (OprgaG). 


a and b are integers, but b is not = to 0, and the operation multiplication, denoted by "x," is (Q,x) a GrouP? 
a and b are integers, but b is not = to 0, and the operation multiplication, denoted by "x," is (Q,x) a GrouP? 
Groups are important, too, because they are a fundamental algebraic structure. We can investigate groups with added properties like CommutativeGroups? and NonCommutativeGroups?, and also build other structures based on the notion of a group, but with more than one operations and more properties. This brings us to RinGs and FielDs. 
Groups are important as a fundamental algebraic structure and as a basic tool for embedding the mathematical properties of objects, even properties of groups themselves because one can built groups out of existing ones (see GroupProductS?, NoetherIsomorphismTheoremS?, WreathProduct?, BrauerGroup?, DerivedGroup?, GroupCenter?, GroupCharacter?, FittingSubgroup?, CohomologyGroups?). We can investigate groups with global properties like CommutativeGroups? and NonCommutativeGroups?, SimpleGroups? and SolvableGroups?, FinitelyGeneratedGroups?, AlgebraicGroups?, study how groups can be related via MorphisMs and also build other structures based on the notion of a group, but with more than one operation and/or more properties. This brings us to RinGs, FielDs, ModulEs. 