Describes elements of D_5 using SAGE

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Discussion Overview

The discussion revolves around the elements of the dihedral group \(D_5\), including their representation and how to generate a Cayley table using SAGE. Participants explore various ways to express the elements and seek assistance with SAGE functionalities.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that the elements of \(D_5\) can be represented as \(R_0, R_{72}, R_{144}, R_{216}, R_{288}, F_1, F_2, F_3, F_4, F_5\).
  • Others suggest an alternative representation to clarify the relationships among elements, such as \(I, R, R^2, R^3, R^4, F, FR, FR^2, FR^3, FR^4\).
  • One participant shares their experience with SAGE, showing how to define \(D_5\) and list its elements as a permutation group.
  • Another participant expresses interest in obtaining LaTeX output from SAGE but encounters difficulties.
  • Some participants discuss the differences in output between SAGE and CoCalc, which also uses SAGE.

Areas of Agreement / Disagreement

There is no consensus on the best way to represent the elements of \(D_5\) or on the functionality of SAGE for generating LaTeX output. Multiple viewpoints and approaches remain in the discussion.

Contextual Notes

Participants mention limitations in their attempts to obtain LaTeX output from SAGE, indicating potential differences in behavior between SAGE and CoCalc.

karush
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Describes the elements of $D_5$
ok I think the elements of $D_5$ are
$$R_0\quad R_{72} \quad R_{144} \quad R_{216} \quad R_{288} \quad F_1 \quad F_2 \quad F_3 \quad F_4 \quad F_5$$

ok from this was going to make a cayley table
but was wondering if anybody know how to do this with SAGE

otherwise I presume we could just create it with latex table
 

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karush said:
Describes the elements of $D_5$
ok I think the elements of $D_5$ are
$$R_0\quad R_{72} \quad R_{144} \quad R_{216} \quad R_{288} \quad F_1 \quad F_2 \quad F_3 \quad F_4 \quad F_5$$

ok from this was going to make a cayley table
but was wondering if anybody know how to do this with SAGE

otherwise I presume we could just create it with latex table

Usually we write the elements such that their relation to the other elements is clear.
And also that we can easily distinguish the ones with different characteristics.

We might write it as:
$$I\quad R \quad R^2 \quad R^3 \quad R^4 \quad F \quad FR \quad FR^2 \quad FR^3 \quad FR^4$$

Just for fun I installed SageMath to see what it does:

sage:[/color] D_5 = DihedralGroup(5)
sage:[/color] D_5
Dihedral group of order 10 as a permutation group

sage:[/color] D_5.list()
[(),
(1,5,4,3,2),
(1,4,2,5,3),
(1,3,5,2,4),
(1,2,3,4,5),
(2,5)(3,4),
(1,5)(2,4),
(1,4)(2,3),
(1,3)(4,5),
(1,2)(3,5)]

sage:[/color] D_5.cayley_table()
* a b c d e f g h i j
+--------------------
a| a b c d e f g h i j
b| b c d e a j f g h i
c| c d e a b i j f g h
d| d e a b c h i j f g
e| e a b c d g h i j f
f| f g h i j a b c d e
g| g h i j f e a b c d
h| h i j f g d e a b c
i| i j f g h c d e a b
j| j f g h i b c d e a
​
 
wow, that is great help

I am going to dive into it a lot more tommorro

I was curious there is a way to get SAGE to output in latex I tried \latex but no
already spent about an hour trying to this but no success
I noticed yours is screenshot

oh this is going to be our new rail access from the campus
 
Last edited:
Klaas van Aarsen said:
Usually we write the elements such that their relation to the other elements is clear.
And also that we can easily distinguish the ones with different characteristics.

We might write it as:
$$I\quad R \quad R^2 \quad R^3 \quad R^4 \quad F \quad FR \quad FR^2 \quad FR^3 \quad FR^4$$

Just for fun I installed SageMath to see what it does:
sage: D_5 = DihedralGroup(5)
sage: D_5
Dihedral group of order 10 as a permutation group

sage: D_5.list()
[(),
(1,5,4,3,2),
(1,4,2,5,3),
(1,3,5,2,4),
(1,2,3,4,5),
(2,5)(3,4),
(1,5)(2,4),
(1,4)(2,3),
(1,3)(4,5),
(1,2)(3,5)]

sage: D_5.cayley_table()
* a b c d e f g h i j
+--------------------
a| a b c d e f g h i j
b| b c d e a j f g h i
c| c d e a b i j f g h
d| d e a b c h i j f g
e| e a b c d g h i j f
f| f g h i j a b c d e
g| g h i j f e a b c d
h| h i j f g d e a b c
i| i j f g h c d e a b
j| j f g h i b c d e a
​


ok done in latex but want to see if SAGE can return this
it may not be an align tho

D_5=DihedralGroup(5)
D_5
D_5.list()
\begin{align*}
&[().\\
&(1,5,4,3,2),\\
&(1,4,2,5,3)\\
&(1,3,5,2,4)\\
&(1,2,3,4,5)\\
&(2,5)(3,4)\\
&(1,5)(2,4)\\
&(1,4)(2,3)\\
&(1,2)(3,5)]\\
\end{align*}
 
I get:

sage:[/color] latex(D_5.list())
$$\left[, (1,5,4,3,2), (1,4,2,5,3), (1,3,5,2,4), (1,2,3,4,5), (2,5)(3,4), (1,5)(2,4), (1,4)(2,3), (1,3)(4,5), (1,2)(3,5)\right]$$​
 
https://www.physicsforums.com/attachments/8594
ok I did this to get the latex output
there might be a deference in SAGE and cocalc which uses SAGE
 
https://dl.orangedox.com/GXEVNm73NxaGC9F7Cy

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