1
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1answer
29 views

Rotation-Reflection-Symmetries

I have the following exercise: Do a rotation of $2 \cdot 72 ^\circ$. Then do a reflection of the axis $d4$. Then do a reflection of the axis $d3$. Then do a rotation of $2 \cdot 72 ^\circ$. ...
0
votes
1answer
49 views

Rotation and Symmetries of Equilateral Triangle

Let $\Sigma=\{ 1,2,3 \}$ be the set of the vertices of an equilateral triangle. Let $f=\sigma$ be the rotation of level with center of rotation $O$ over an angle of $\frac{2 \pi}{3}$ radians or ...
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votes
1answer
69 views

Symmetries of a square and it's similarity to the Division Algorithm

I need help with this question: (each variable $r$ represents a rotation of the square about the axis through its centroid at $90^{\circ}$ intervals. $e$ represents nonmotion. This question is ...
2
votes
1answer
65 views

Surjectivity for permutation representation of a group action

I am having trouble proving that my function is surjective. Here is the problem statement: Problem statement: Let T be the tetrahedral rotation group. Use a suitable action of T on some set, and the ...
0
votes
1answer
131 views

Conjugacy classes of rotational symmetry tetrahedron

I am struggling with this question, as I don't know how to give a "short proof" as requested by our professor, as a non-graded exercise. Especially, since we have to do this without using any shapes, ...
3
votes
1answer
206 views

Decomposition formulas for rotational symmetries of a cube

I have a problem that I would like to check my work on. I am also stuck on the verifications for $E$ and $F$. Any help would be greatly appreciated. Thanks in advance. Problem statement: Let $G$ be ...
1
vote
2answers
125 views

Using counting formula to get |G| = |kernel φ||image φ|

The counting formula I am saying : Let S be a finite set on which a group G operates, and let Gs and Os be the stabilizer and orbit of an element s of S. Then |G|=|Gs||Os| or ...
1
vote
0answers
272 views

a conjugate of a glide reflection by any isometry of the plane is again a glide reflection

Q : Prove that a conjugate of a glide reflection by any isometry of the plane is again a glide reflection, and show that the glide vectors have the same length. I know that: the conjugate ...
1
vote
2answers
127 views

Frieze groups, wallpaper groups

Can someone suggest a source that proves the classifications of the 7 frieze groups and 17 wallpaper groups in an elegant way?
2
votes
3answers
429 views

Symmetry group of a triangular lattice

The question is: Let $G$ be the group of symmetries of an equilateral triangular lattice $L$. Find the index in $G$ of the subgroup $T \cap G$. ($T$ is the group of translations) The solution ...
0
votes
1answer
189 views

Algebra - abstract symmetry - counting

I am reading Michael Artin's Algebra, encountered paragraphs( 7.3) that I don't quite understand. so there's the citation The elementary formula which uses the partition of $S$ into orbits to ...
10
votes
5answers
411 views

Definition of group action

I'm currently taking a class in abstract algebra, and the textbook we are using is Ted Shifrin's Abstract Algebra: A Geometric Approach. In the chapter on group actions and symmetry, he defines a ...