The unit group modulo one hundred is isomorphic to C_20 times C_2

Given: The objects and hypotheses in the example.

Let $n\ge 1$ be an integer. Multiplication makes $\mathbb{Z}\mathrm{/}n$ a commutative monoid with identity $[1{]}_n$ by thm-integers-modulo-n-basic-algebra. A class $u\in \mathbb{Z}\mathrm{/}n$ is a unit when it is invertible in that monoid (def-invertible-element). The set of all units is $(\mathbb{Z}\mathrm{/}n{)}^{\times }:=\{u\in \mathbb{Z}\mathrm{/}n:\text{some }v\in \mathbb{Z}\mathrm{/}n\text{ satisfies }uv=[1{]}_n\}\mathrm{.}$ By lem-monoid-units-form-a-group, multiplication restricts to a group operation on $(\mathbb{Z}\mathrm{/}n{)}^{\times }$, called the unit group modulo $n$. The quotient $\mathbb{Z}\mathrm{/}n$ is finite with cardinality $n$ by thm-standard-representatives-modulo-n, and its unit set is a finite subset by thm-subset-of-a-finite-set. Euler's totient function is therefore defined for every positive integer $n$ by $\phi (n):=\mid (\mathbb{Z}\mathrm{/}n{)}^{\times }\mid \in \mathbb{N}$ (def-finite-cardinality). For $n=1$, the quotient has one element, which is its multiplicative identity and hence a unit, so $\phi (1)=1$ follows from the definition. (The unit group $(\mathbb{Z}\mathrm{/}n{)}^{\times }$ and Euler's totient $\phi (n)=\mid (\mathbb{Z}\mathrm{/}n{)}^{\times }\mid$ for $n\ge 1$).

Let $G$ be a group and let $N_0,\dots ,N_{r-1}$ be normal subgroups, where $r\in \mathbb{N}$. They form an internal direct product when they generate $G$ and, for each $i<r$, $N_i\cap ⟨N_j:j<r,j\mathrm{\ne }i⟩=\{e\}\mathrm{.}$ The empty family is an internal direct product of the trivial group. For two subgroups of an abelian group this says $G=HK$ and $H\cap K=\{e\}$; in additive notation one writes $G=H\oplus K$. Normal subgroups and generated subgroups are those of def-normal-subgroup and def-generated-subgroup, and the comparison product is def-external-direct-product-of-groups. (Internal direct products of finitely many normal subgroups).

Let $N_0,\dots ,N_{r-1}⊴G$. The following are equivalent: the $N_i$ form an internal direct product of $G$; every $g\in G$ has a unique expression $g=n_0\cdots n_{r-1}$ with $n_i\in N_i$; and the multiplication map $\mu :{\prod }_{i<r}{N}_i\to G$ is an isomorphism. These statements include the empty family and the one-factor case. (Internal direct products are external direct products, equivalently every element has a unique factorisation).

For every finite abelian group $G$ there is a unique list $1<n_1\mid \cdots \mid n_r$ such that $G\cong C_{n_1}\times \cdots \times C_{n_r}$. Moreover $\mathrm{\mid }G\mathrm{\mid }=n_1\cdots n_r$. The trivial group corresponds to the empty list and empty product. (Fundamental theorem of finite abelian groups: invariant-factor form).