A [[Trivial]] [[Group]] (or *zero group*) is a [[Group]] whose [[Set|set]] $C$ has only one element. All [[Trivial Group|Trivial Groups]] are [[Isomorphism|Isomorphic]] with each other. A [[Trivial Group]] is always a [[Cyclic Group]] of order $1$, therefore it can be denoted as $C_1$. By definition of a group, any [[Trivial Group]]'s only member is the [[Identity Function|identity]] $e$. $\huge C = \set{e} $ For any [[Group]] $(G, \circ)$, there always exists at least one [[Subgroup]] of $G$ who has a single element that is the [[Identity Function|identity]] of $G$ $\op{id_{(G,\circ)}}$, which is always a [[Trivial Group]]. When discussing an [[Additive Group]], its [[Trivial Group]] is often denoted as $0$, while when discussing a [[Multiplicative Group]] it is denoted as $1$. >[!info] >Found a really funny example where when you are talking about the [[Trivial Ring]], addition and multiplication are identical, therefore you can make this statement about the trivial groups for multiplication & addition for that [[Ring]]: >$ 0 = 1 $