Understanding Symmetry Groups of a Pentagon

[Natasha1]

Right I have been asked to identify the elements of a full symmetry group of a pentagon and give the order of each element?

I have found the elements which I have marked as I, R (Rotation of 72 degrees), R2 (Rotation of 144 degrees), R3 (Rotation of 216 degrees), R4 (Rotation of 288 degrees), M (reflection in axis 1), MR (transformation from rotation R and reflection M), MR2, MR3, MR4.

Now I have made the cayley table of the full symmetry group of the pentagon using these elements.

But what is the order of each element?

What are the subgroups and why?

Can I generate the whole group with two reflections? Any two reflections?

Please help me as I need the answers for tomorrow :(

 

[marlon]

Right I have been asked to identify the elements of a full symmetry group of a pentagon and give the order of each element?
I have found the elements which I have marked as I, R (Rotation of 72 degrees), R^2 (Rotation of 144 degrees), R^3 (Rotation of 216 degrees), R4 (Rotation of 288 degrees), M (reflection in axis 1), MR (transformation from rotation R and reflection M), MR^2, MR^3, MR^4.
Now I have made the cayley table of the full symmetry group of the pentagon using these elements.

But what is the order of each element?
What are the subgroups and why?
Can I generate the whole group with two reflections? Any two reflections?
Please help me as I need the answers for tomorrow :(

Ok, sorry but what exactly is your background in this ? I mean, do you know how to calculate a Cayley Table ?

I also suggest you first revise your intro group theory, because otherwise there is no point in making this exercise.

marlon

 

[Natasha1]

That's really useful thanks! I mean come on. On don't need to be told to go and revise, please.

Anyone else?

 

[Natasha1]

My Cayley table looks like this...

I R R2 R3 R4 M MR MR2 MR3 MR4
I I R R2 R3 R4 M MR MR2 MR3 MR4
R R R2 R3 R4 I MR2 MR3 MR4 M MR
R2 R2 R3 R4 I R MR4 M MR MR2 MR3
R3 R3 R4 I R R2 MR MR2 MR3 MR4 M
R4 R4 I R R2 R3 MR3 MR4 M MR MR2
M M MR3 MR MR4 MR2 I R2 R4 R R3
MR MR MR4 MR2 M MR3 R3 I R2 R4 R
MR2 MR2 M MR3 MR MR4 R R3 I R2 R4
MR3 MR3 MR MR4 MR2 M R4 R R3 I R2
MR4 MR4 MR2 M MR3 MR R2 R4 R R3 I

 

[Natasha1]

Someone out there please?

 

[matt grime]

Marlon is perfectly correct to tell you to revise: if you don't know what "the order of an element" means then you can't answer the question. If you do know what it means then it is easy to work out either from geometric intuition or from the cayley table.

So, how many times do you apply the basic rotation before you get the identity? Or any other rotation? Or any reflection?

The subgroup question is straightforward again fi you know what Lagrange's theorem states, as is the last question. So, go and revise one fact: If H is a subgroup of G then the order of H INSERT WORD HERE the order of G