Intereting Posts

Rings with $a^5=a$ are commutative
Namesake of Cantor's diagonal argument
Why are linear functions linear?
Is it possible to assign a value to the sum of primes?
Importance of rank of a matrix
proving the inequality $\triangle\leq \frac{1}{4}\sqrt{(a+b+c)\cdot abc}$
Why don't we have an isomorphism between $R$ and $ R]$?
How to show that restricted Lorentz group (orthochoronous proper Lorentz transformations) is a normal subgroup?
Provide different proofs for the following equality: $\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$
Prisoners Problem
Fast calculation for $\int_{0}^{\infty}\frac{\log x}{x^2+1}dx=0$
Proof for exact differential equations shortcut?
Exchange integral and conditional expectation
$19$ divides $ax+by+cz$
How to find a vector potential (inverse curl)?

It seems that $\{e\}, \{e,s\}, \{e,rs\}, \{e,r^2s\},…,\{e,r^{n-1}s\}, \{e,r,r^2,…,r^{n-1}\}, D_n$ are always subgroups of $D_n$.

Especially when $n$ is odd, these seem to be the only subgroups.

But when n is even, say $n=4$, then there are also $\{e,s,r^2,r^r2s\}$ and $\{e,rs,r^2,r^3s\}$.

- In a finite abelian group of order n, must there be an element of order n?
- Characterizing all ring homomorphisms $C\to\mathbb{R}$.
- Does $S = R \cap K$ of a field extension $K \subseteq L = Q(R)$ satisfy $Q(S) = K$?
- Show that the set of functions under composition is isomorphic to $S_3$
- The (Jacobson) radical of modules over commutative rings
- Unique factorization domain that is not a Principal ideal domain

It makes me wonder, is there a general formula/algorithm for finding all subgroups of $D_n$ when $n$ is even?

- Given fields $K\subseteq L$, why does $f,g$ relatively prime in $K$ imply relatively prime in $L$?
- Is my proof that $U_{pq}$ is not cyclic if $p$ and $q$ are distinct odd primes correct?
- Show that number of solutions satisfying $x^5=e$ is a multiple of 4?
- Is there a proof of the irrationality of $\sqrt{2}$ that involves modular arithmetic?
- Localization of ideals at all primes
- Different ways of constructing the free group over a set.
- What are central automorphisms used for?
- Question about solvable groups
- The group of $k$-automorphisms of $k]$, $k$ is a field
- Variety generated by finite fields

Yes, there is a general classification of all subgroups of $D_n$ for every $n$.

**Theorem**: Every subgroup of $D_n=\langle r,s \rangle$ is is either cyclic or dihedral, and a complete listing of the subgroups is as follows:

(1) $\langle r^d\rangle$, where $d\mid n$, with index $2d$,

(2) $\langle r^d, r^is \rangle$, where $d\mid n$, $0\le i\le d-1$, with index $d$.

Every subgroup of $D_n$ occurs exactly once in this listing.

For a proof see Theorem 3.1 of Keith Conrad’s notes. Furthermore the cases $n$ even and $n$ odd are discussed in more detail – see section $3$.

- Number of elements of order $7$ in a group
- How does a complex power series behave on the boundary of the disc of convergence?
- The product of n consecutive integers is divisible by n factorial
- If every composition of a differentiable path and a function is differentiable at 0, means the function is differentiable at 0
- Is this Epsilon-Delta approach to prove that $e^x$ is continuous correct?
- Dirichlet's theorem on arithmetic progressions
- Poincaré Inequality
- What is $0\div0\cdot0$?
- Proof of the Usual Topology on R
- Does the recursive sequence $a_1 = 1, a_n = a_{n-1}+\frac{1}{a_{n-1}}$ converge?
- Show that $f^{(n)}(0)=0$ for $n=0,1,2, \dots$
- Is a bra the adjoint of a ket?
- Simplifying my sum which contains binomials
- Proof of the Banachâ€“Alaoglu theorem
- If $\dfrac{a}{b-c}+\dfrac{b}{c-a}+\dfrac{c}{a-b}=0$ with unequal $a,b,c$, Prove that $\dfrac{a}{(b-c)^2}+\dfrac{b}{(c-a)^2}+\dfrac{c}{(a-b)^2}=0$