Trivial group - Citizendia

In mathematics, a trivial group is a group consisting of a single element. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element All such groups are isomorphic so one often speaks of the trivial group. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective The single element of the trivial group, variously labeled e, 1, or 0, is the identity element. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that The group operation is e + e = e.

Every trivial group is abelian and cyclic; all these results being trivial, hence the name. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an The trivial group is often written as Z₁ or just 0, 1, or e.

The subgroup of a group G consisting of just the identity element is called the trivial subgroup of G. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of It is, of course, a trivial group.

The trivial group should not be confused with the empty set (which has no elements and therefore, lacking an identity element, cannot be a group). In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members The trivial group serves as the zero object in the category of groups. In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such (The category of sets, on the other hand, has no zero objects; the empty set only serves as an initial object while any singleton set serves as a terminal object). In Mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose Morphisms are In Mathematics, a singleton is a set with exactly one element

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