Trivial group

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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a trivial group or zero group is a group consisting of a single element. All such groups are
isomorphic In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
, so one often speaks of the trivial group. The single element of the trivial group is the
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a Set (mathematics), set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in alge ...
and so it is usually denoted as such: $0, 1,$ or $e$ depending on the context. If the group operation is denoted $\, \cdot \,$ then it is defined by $e \cdot e = e.$ The similarly defined is also a group since its only element is its own inverse, and is hence the same as the trivial group. The trivial group is distinct from the
empty set In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
, which has no elements, hence lacks an identity element, and so cannot be a group.

# Definitions

Given any group $G,$ the group consisting of only the identity element is a
subgroup In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ...
of $G,$ and, being the trivial group, is called the of $G.$ The term, when referred to "$G$ has no nontrivial proper subgroups" refers to the only subgroups of $G$ being the trivial group $\$ and the group $G$ itself.

# Properties

The trivial group is cyclic of order $1$; as such it may be denoted $\mathrm_1$ or $\mathrm_1.$ If the group operation is called addition, the trivial group is usually denoted by $0.$ If the group operation is called multiplication then 1 can be a notation for the trivial group. Combining these leads to the
trivial ring In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique Ring (mathematics), ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any Rng (algebra)#Rng of square z ...
in which the addition and multiplication operations are identical and $0 = 1.$ The trivial group serves as the zero object in the
category of groups In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
, meaning it is both an
initial object In category theory, a branch of mathematics, an initial object of a category (mathematics), category is an object in such that for every object in , there exists precisely one morphism . The dual (category theory), dual notion is that of a t ...
and a terminal object. The trivial group can be made a (bi-) ordered group by equipping it with the trivial non-strict order $\,\leq.$