Let $M$ be an additive subgroup of $\mathbf { Z } ^ { n }$ with $n \in \mathbf { N }$ (we agree that $\mathbf { Z } ^ { 0 }$ is the trivial group). We propose to prove by induction on $n$ the following result: (*) There exists $r \in \mathbf { N }$ such that the abelian group $M$ is isomorphic to $\mathbf { Z } ^ { r }$. a) Verify the cases $n = 0$ and $n = 1$. We now assume the result is true for $n - 1$. Let $p : \mathbf { Z } ^ { n } \rightarrow \mathbf { Z }$ be the projection onto the first coordinate, we denote by $N$ the kernel of $p$ and $N _ { 1 } = M \cap N$, then we set $p ( M ) = a \mathbf { Z }$ with $a \in \mathbf { Z }$. We choose $e _ { 1 } \in M$ such that $p \left( e _ { 1 } \right) = a$. Show that if $a \neq 0$, then the application $$N _ { 1 } \times \mathbf { Z } \rightarrow M , ( x , m ) \mapsto x + m e _ { 1 }$$ is a group isomorphism. b) Deduce (*). c) Show that the integer $r$ such that $M$ is isomorphic to $\mathbf { Z } ^ { r }$ is unique (one may consider the rank of a family of vectors of $\mathbf { Z } ^ { r }$ in the $\mathbf { Q }$-vector space $\mathbf { Q } ^ { r }$).
Let $M$ be an additive subgroup of $\mathbf { Z } ^ { n }$ with $n \in \mathbf { N }$ (we agree that $\mathbf { Z } ^ { 0 }$ is the trivial group). We propose to prove by induction on $n$ the following result:\\
(*) There exists $r \in \mathbf { N }$ such that the abelian group $M$ is isomorphic to $\mathbf { Z } ^ { r }$.\\
a) Verify the cases $n = 0$ and $n = 1$. We now assume the result is true for $n - 1$. Let $p : \mathbf { Z } ^ { n } \rightarrow \mathbf { Z }$ be the projection onto the first coordinate, we denote by $N$ the kernel of $p$ and $N _ { 1 } = M \cap N$, then we set $p ( M ) = a \mathbf { Z }$ with $a \in \mathbf { Z }$. We choose $e _ { 1 } \in M$ such that $p \left( e _ { 1 } \right) = a$. Show that if $a \neq 0$, then the application
$$N _ { 1 } \times \mathbf { Z } \rightarrow M , ( x , m ) \mapsto x + m e _ { 1 }$$
is a group isomorphism.\\
b) Deduce (*).\\
c) Show that the integer $r$ such that $M$ is isomorphic to $\mathbf { Z } ^ { r }$ is unique (one may consider the rank of a family of vectors of $\mathbf { Z } ^ { r }$ in the $\mathbf { Q }$-vector space $\mathbf { Q } ^ { r }$).