As shown in this note, the symmetry group $S_4$ for a cube has $3$ subgroups that are isomorphic to $D_4$, the dihedral group of order $2 \times 4 = 8$. How to geometrically illustrate this fact? Specifically, where are the squares embedded in the cube? Related post: How to geometrically show that there are $4$ […]

Name $r$ = clockwise 90 deg. rotation and $f$ = flip across the square’s vertical axis = the brown $\color{brown}{f}$ in my picture underneath. Zev Chonoles’s $f$ is different. Carter fleshes out why $frf = r^{-1} $ intuitively: (1.) Can someone please unfold, like Carter, why $fr = r^{-1}f $? I see why for this […]

Let $G$ be the set of bijections $\mathbb{R} \to \mathbb{R}$ which preserve the distance between pairs of points, and send integers to integers. Then $G$ is a group under composition of functions. The following two elements are obviously in $G$: the function $t$ (translation) where $t(x)=x+1$ for each $x \in \mathbb{R}$ and the function $r$ […]

I have been thinking about a composition series for $D_{14}\times D_{10}$ (where $D_{2n}$ is the dihedral group with $2n$ elements). Is the following a correct composition series for $D_{10}\times D_{14}$: $$D_{14}\times D_{10}\vartriangleright \langle\sigma_1\rangle\times D_{10}\vartriangleright\langle\sigma_1\rangle\times\langle\sigma_2\rangle\vartriangleright\{id_1\}\times\{id_2\}?$$ I also have to verify what the factors are, and in this case, I think the factors are $$\begin{align}&(D_{14}\times D_{10})/(\sigma_1\times D_{10})\cong\mathbb{Z}_2\\& […]

$G=D_6$ and $H=<R^2>$. Use this Cayley table for $D_6$ (a). Show that $H \vartriangleleft G$. I want to show by finding out $aH=Ha$ for all $a \in G$, but then how do I proceed, it would be too tedious to check all $a$ in $D_6$, is there any way else to show it? (b). List […]

I was wondering if there is a special technique to find the conjugacy classes of $\mathcal D_{10}=\left<a,b\mid a^5=b^2=1,bab^{-1}=a^{-1}\right>$, and of $\mathcal D_{2n}=\left<a,b\mid a^n=b^2=1,bab^{-1}=a^{-1}\right> $ ? I can do it by hand (but it’s a little bit long), and there is probably a simple technic.

I am reading Abstract Algebra, Theory and Applications by Judson and in exercise $13$ chapter $9$, Isomorphisms, I need to prove that the set of matrices $$A=\pmatrix{ \omega & 0 \\ 0 & \omega ^{-1}} \qquad B=\pmatrix{ 0 & 1 \\ 1 & 0}$$ Where $\omega = e^{2\pi i /n}$ form a group isomorphic to […]

In an proof that I recently read, the following ‘fact’ is used, where $D_{2n}$ denotes the dihedral group of order $2n$: If $n$ is even, then $D_{2n} \cong C_2 \times D_n$. The (short) given justification is that the centre $Z(D_{2n}) \cong C_2$, whenever $n$ is even, and is trivial provided that $n$ is odd. However, […]

How to prove that the center of the dihedral group $D_{2n}$ is $\{1,r^{n}\}$ and the center of $D_{2n-1}$ is $\{1\}$? I don’t know how to prove it in this general case.

Consider the conjugation action of $D_n$ on $D_n$. Prove that the number of conjugacy classes of the reflections are $\begin{cases} 1 &\text{ if } n=\text{odd} \\ 2 &\text{ if } n=\text{even} \end{cases} $ I tried this: Let $σ$ be a reflection. And $ρ$ be the standard rotation of $D_n$. $$ρ^l⋅σρ^k⋅ρ^{-l}=σρ^{k-2l}$$ $$σρ^l⋅σr^k⋅ρ^{-l}σ=σρ^{-k+2l}$$ If $n$ is even, […]

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