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In abstract algebra, a group is a set with a binary operation that satisfies certain axioms, detailed below. For example, the set of integers with addition is a group. The branch of mathematics which studies groups is called group theory. Many of the structures investigated in mathematics turn out to be groups. These include familiar number systems, such as the integers, the rational numbers, the real numbers, and the complex numbers under addition, as well as the non-zero rationals, reals, and complex numbers, under multiplication. Other important examples are the group of non-singular matrices under multiplication and the group of invertible functions under composition. Group theory allows for the properties of such structures to be investigated in a general setting. Group theory has extensive applications in mathematics, science, and engineering. Many algebraic structures such as fields and vector spaces may be defined concisely in terms of groups, and group theory provides an important tool for studying symmetry, since the symmetries of any object form a group. Groups are thus essential abstractions in branches of physics involving symmetry principles, such as relativity, quantum mechanics, and particle physics. Furthermore, their ability to represent geometric transformations finds applications in chemistry, computer graphics, and other fields.

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Q: How can you Show that the set of integers is a group with respect to addition?
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Show that the set of integers with respect to multiplication is not a group?

To be a group, the set of integers with multiplication has to satisfy certain axioms: - Associativity: for all integers x,y and z: x(yz) = (xy)z - Identity element: there exists some integer e such that for all integers x: ex=xe=x - Inverse elements: for every integer x, there exists an integer y such that xy=yx=e, where e is the identity element The associativity is satisfied and 1 is clearly the identity element, however no integer other than 1 has an inverse as in the integers xy = 1 implies x=y=1

Does 0 and -1 show closure property with respect to addition?


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