# Describes elements of D_5 using SAGE

• MHB
• karush
In summary: I am not sure if that is correct, but I am not familiar with SageMath enough to know for sure. As for the cocalc and Sage difference, I am not familiar with cocalc so I can't say for sure.
karush
Gold Member
MHB
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: 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

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
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: 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

## 1. What is SAGE?

SAGE (Systems Approach to Geometric Exploration) is a mathematical software system used for mathematical computation, data analysis, and visualization. It is primarily used for research and educational purposes in mathematics, computer science, and other related fields.

## 2. What are elements in the context of mathematics?

In mathematics, elements refer to the individual objects or members of a set. In the case of D5, elements would refer to the symmetries or transformations of a regular pentagon.

## 3. How does SAGE describe elements of D5?

SAGE has built-in functions and commands specifically designed to work with groups, including D5. These functions can describe the elements of D5 by showing their properties, such as rotations and reflections.

## 4. What is the significance of D5 in mathematics?

D5 is a finite group that represents the symmetries of a regular pentagon. It is significant in mathematics as it is one of the five Platonic groups, which are important in the study of geometric symmetries. It also has applications in crystallography and the study of molecular structures.

## 5. How can SAGE be used to study D5?

SAGE can be used to explore and analyze the properties and operations of D5, such as finding the order of the group, determining its subgroups, and calculating its Cayley table. It can also be used to visualize the symmetries of a regular pentagon and demonstrate how they relate to the elements of D5.