Trivial group

In mathematics, a trivial group or zero group is a group that consists of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually denoted as such: ⁠

0

{\displaystyle 0}

⁠, ⁠

1

{\displaystyle 1}

⁠, or ⁠

e

{\displaystyle \mathrm {e} }

⁠ depending on the context. If the group operation is denoted ⁠

⋅

{\displaystyle \,\cdot \,}

⁠ then it is defined by ⁠

e

⋅

e

=

e

{\displaystyle \mathrm {e} \cdot \mathrm {e} =\mathrm {e} }

⁠.

The similarly defined trivial monoid 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, which has no elements, hence lacks an identity element, and so cannot be a group.

Definitions

Given any group ⁠

G

{\displaystyle G}

⁠, the group that consists of only the identity element is a subgroup of ⁠

G

{\displaystyle G}

⁠, and, being the trivial group, is called the trivial subgroup of ⁠

G

{\displaystyle G}

⁠.

The term, when referred to "⁠

G

{\displaystyle G}

⁠ has no nontrivial proper subgroups" refers to the only subgroups of ⁠

G

{\displaystyle G}

⁠ being the trivial group ⁠

{

e

}

{\displaystyle \{\mathrm {e} \}}

⁠ and the group ⁠

G

{\displaystyle G}

⁠ itself.

Properties

The trivial group is cyclic of order ⁠

1

{\displaystyle 1}

⁠; as such it may be denoted ⁠

Z

1

{\displaystyle \mathrm {Z} _{1}}

⁠ or ⁠

C

1

{\displaystyle \mathrm {C} _{1}}

⁠. If the group operation is called addition, the trivial group is usually denoted by ⁠

0

{\displaystyle 0}

⁠.