\(
\newcommand{\cat}[1] {\mathrm{#1}}
\newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})}
\newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}}
\newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}}
\newcommand{\betaReduction}[0] {\rightarrow_{\beta}}
\newcommand{\betaEq}[0] {=_{\beta}}
\newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}}
\newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}}
\newcommand{\groupMul}[1] { \cdot_{\small{#1}}}
\newcommand{\groupAdd}[1] { +_{\small{#1}}}
\newcommand{\inv}[1] {#1^{-1} }
\newcommand{\bm}[1] { \boldsymbol{#1} }
\require{physics}
\require{ams}
\require{mathtools}
\)
Math and science::Algebra::Aluffi
Subgroups of cyclic groups
The following two propositions determine every subgroup of every cyclic
group.
Subgroup of \( \mathbb{Z} \)
Let \( G \) be a subgroup of \( \mathbb{Z} \). Then [\( G = \; ? \)]
for some \( d \geq 0 \).
Subgroup of \( \mathbb{Z}/n\mathbb{Z} \)
Let \( G \) be a subgroup of \( \mathbb{Z}/n\mathbb{Z} \) for some
\( n \geq 0 \). Then [\( G = \; ? \)] for some
\( d \geq 0 \) with [what condition?].
Can you recall the proofs?
