Trivial group

In

and so it is usually denoted as such: ${\displaystyle 0,1,}$ or ${\displaystyle e}$ depending on the context. If the group operation is denoted ${\displaystyle \,\cdot \,}$ then it is defined by ${\displaystyle e\cdot e=e.}$

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 ${\displaystyle G,}$ the group consisting of only the identity element is a subgroup of ${\displaystyle G,}$ and, being the trivial group, is called the trivial subgroup of ${\displaystyle G.}$

The term, when referred to "${\displaystyle G}$ has no nontrivial proper subgroups" refers to the only subgroups of ${\displaystyle G}$ being the trivial group ${\displaystyle \{e\}}$ and the group ${\displaystyle G}$ itself.

Properties

The trivial group is cyclic of order ${\displaystyle 1}$; as such it may be denoted ${\displaystyle \mathrm {Z} _{1}}$ or ${\displaystyle \mathrm {C} _{1}.}$ If the group operation is called addition, the trivial group is usually denoted by ${\displaystyle 0.}$ If the group operation is called multiplication then 1 can be a notation for the trivial group. Combining these leads to the

${\displaystyle 0=1.}$

The trivial group serves as the

The trivial group can be made a (bi-)

${\displaystyle \,\leq .}$

See also

• Zero object (algebra) – Algebraic structure with only one element
• List of small groups

References

• Rowland, Todd & Weisstein, Eric W. "Trivial Group". MathWorld.