[Math] The order of $R^k \in D_n$, the $n$th dihedral group

abstract-algebradihedral-groupsgroup-theory

I am using Thomas Judson's open-sourced Abstract Algebra textbook, available at http://abstract.ups.edu/aata, and I'm having trouble with this problem:

Let $R$ represent a rotation element in $D_n$, the $n$th dihedral group. Prove that the order of $R^k \in D_n$ is $n/\gcd(n,k)$.

Here's what I've done/know so far:

  • Let $n,k \in \mathbb{N}$. Treat $k$ as if it were under (mod $n$);
    i.e. if $R^\kappa \in D_n$ such that $\kappa > n$, then rewrite
    $\kappa$ as $\kappa \equiv k (\text{mod} \; n)$. Otherwise, we
    already have that $R^k \in D_n$ such that $1 \leq k \leq n$.
  • Because $R \in D_n$, $R^n=1$ means $R$ is a counter-clockwise
    rotation of $2\pi/n$ radians, so $R^k$ would be a counter-clockwise
    rotation of $2\pi k/n$ radians.
  • For any arbitrary element $a$ in a group $G$, the order of $a$ is the
    smallest positive integer $n$ such that $a^n=1$. ($e$ is typically
    the identity element for this author; but, he tends to use $e=1$ for
    permutation groups).
  • By definition of greatest common divisor, there exist $a,b \in
    \mathbb{Z}$ such that $ak+bn = d = \gcd(n,k)$.

I don't know how to connect these things to the order of $R^k$.

Note: I don't know if any of these concepts are relevant, but references to these aspects of group theory are not (yet) developed for this author's course structure: cosets, Lagrange's theorem, Fermat's and Euler's theorems, isomorphisms, direct products, factor groups, normal subgroups, alternating groups, homomorphisms, automorphisms, Burnside's counting theorem, the class equation, and Sylow theorems. I'll get to those ideas in later chapters.

Best Answer

Since $R$ generates a cyclic subgroup $C_n$ of $D_n$, the element $R^k$ has order $n/gcd(n,k)$, because this holds in general for cyclic groups of order $n$, see here for a proof.

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