TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a trivial group or zero group is a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
consisting of a single element. All such groups are
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, so one often speaks of the trivial group. The single element of the trivial group is the
identity element In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
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 should not be confused with the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

, which has no elements, and lacking an identity element, 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 Cycle or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in social scienc ...

of order $1$; as such it may be denoted $Z_1$ or $C_1.$ 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 zer ...
in which the addition and multiplication operations are identical and $0 = 1.$ The trivial group serves as the
zero object In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
in the
category of groups In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, meaning it is both an
initial object In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
and a
terminal object In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
. The trivial group can be made a (bi-)
ordered group In abstract algebra, a partially ordered group is a group (''G'', +) equipped with a partial order Image:Hasse diagram of powerset of 3.svg, 250px, The Hasse diagram of the power set, set of all subsets of a three-element set , ordered by inclus ...
by equipping it with the trivial
non-strict order 250px, The set of all subsets of a three-element set , ordered by inclusion. Distinct sets on the same horizontal level are incomparable with each other. Some other pairs, such as and , are also incomparable. In mathematics, especially order the ...
$\,\leq.$