$D_n$的子群个数为$d(n)+σ(n)$

hbghlyj

en.wikipedia.org/wiki/Dihedral_group

the total number of subgroups of dihedral group $D_n(n ≥ 1)$, is equal to $d(n) + σ(n)$, where $d(n)$ is the number of positive divisors of $n$ and $σ(n)$ is the sum of the positive divisors of $n$.

若$m|n$,则$D_n$的一个子群$⟨r^{n/m}⟩≌\Bbb Z_m$.当$m$取遍$n$的约数时,同构于$\Bbb Z_m$的子群共有$d(n)$个.
若$m|n$,则$D_n$的$\frac nm$个子群$⟨r^{n/m},r^ks⟩≌D_m$,其中$k=0,1,⋯,\frac nm$.当$m$取遍$n$的约数时,同构于$D_m$的子群共有$σ(n)$个.