Subgroups Of The Dihedral Group

Students in abstract algebra and number theory are usually interested to see that the arithmetic functions $\phi(n)$, Euler's function, and $d(n)$, the number of positive divisors of $n$, occur quite naturally in the solution of some group theory problems. It appears that the function $\sigma(n)$, the sum of the positive divisors of $n$, does also, in counting the number of subgroups of the dihedral group $D_n$. A formula is developed here for that number. Theorem. The number of subgroups of the dihedral group $D_n(n \geqq 3)$ is $d(n)+\sigma(n)$. Proof. When considered geometrically, $D_n$ consists of $n$ rotations and $n$ reflections of the regular $n$-gon. The subgroups of $D_n$ are of two types: (1) Those containing rotations only, and (2) those containing rotations and reflections. The subgroups of type 1 are simply the subgroups of $Z_n$, the cyclic group of order $n$, and the number of them is $d(n)$. The subgroups of type 2 contain an equal number of rotations and reflections, say $t$, of each. Now the $t$ rotations must comprise the unique subgroup of $Z_n$ of order $t$, whence $t \mid n$, but the $t$ reflections can be chosen in several ways. In fact, the axes of reflection form a star-shaped figure with equal central angles which can be positioned in the $n$-gon in $n / t$ ways. Thus for each divisor $t$ of $n$, there are $n / t$ subgroups of type 2 , and the total number of subgroups of type 2 must be $$ \sum_{t \mid n} n / t=\sum_{t \mid n} t=\sigma(n) . $$ This completes the proof of the theorem. It is of interest to note that when $p$ is a prime greater than 2, the number of subgroups of $D_p$ is simply $p+3$. Subgroups of dihedral groups (1)