Order of subgroup G - representing triangular prism

[rohan03]

Please see attached diagram

here is what I have done in order to answer this question
Triangular prism above represents the group G of all symmetries of the prism as permutation of the set {123456}
Part a: is to describe geometrically the symmetries of the prism represented in the cycle for by (14)(26)(35) and (13)(46)
My working out is :
(14)(26)(35) is rotation through pi about axis through the midpoints of edges 14 and plane 2365
And for (13)(46) is the reflection in the plane thorugh the edge 25 and the midpoints of the edges 46 and 13


Part b was to write down all the symmetries of the prism describing geometrically
My working :
• {e} the identity
• (456)(123) rotation through 2pi/3
• (465)(132)- rotation through 4pi/3
• (14)(26)(35) – roation through pi
• (25(16)(34) again rotation through pi –
• (36)(15)(24) rotation through pi-
(I have cut down the description such as centre of the edges and plane etc for all of these just to speed up the writing process)

Then I described indirect symmetries as follows:
• e o (14)(36)(25)= (14) (36) (25)
• (456)(123) o (14) (36) (25) = (153426)
• (465)(132)o(14) (36) (25) = (162435)
• (14) (26) (35) o(14) (26)(35) =(32)(56)
• (36) (15)(24)o(14) (36) (25) = (12) (45)
• (25) (16) (34)o (14) (36) (25) =(13) (46)

I also know there geometric description

Part c is to work out conjugacy classes so this is what I worked out :
• C1 = the identity {e}
• C2- the rotation through pi about the axis through the centre of the edge 14 and lane 2365, the centre of the edge 25 and plane 1365 and edge 36 and the plane 1254 giving
• {(14) (26) (35), (25) (16) (34), (36) (15) (24)}
• C3- rotation through 2pi/3 4pi/3 etc giving
• {(456) (123), (465) (132)}
• C4 – the rflection in the plane through the edges 14 and 56and 23 etc…. giving
{ (32) (56), (12) (45), (13) (46)}
• C5 reflection through vertical PLne giving {(14)(36)(25)}
• C6 is basically
• {(153426) (162435)}
Upto this point I have no problem where I can't progress and don’t understand how to go about it is following part:
Determine a subgroup of G of order2, a subgroup of order 4 and subgroup of order 6. And deciding if the subgroup is normal or not.

Now I have a slight idea but what I fail to understand is how to workout number of elemnts in each conjugacy classes and the order of this subgroup – I guess this is a S3 so order of subgroup must divide by 12 by Lagrange’s theorem but this is just a guess and if I am on right line- I still havent’ got a clue how to go about finding this out. So please help.

 

[suziequeweb]

Where is your diagram?

 

[rohan03]

hi this question is resolved- I have posted another question under homework section under calculus - riemann integral- can you please look at it? Thanks

 

[suziequeweb]

I can have a look, can you post diagram for this one also though as a student of mine is also looking for it and it may be helpful

 

[rohan03]

Hi May I ask what do you do? Seems like you are a teacher- so it should be pretty straightforward for you to sketch one for your student.

I will have a look but chances are slim as I delete every document that I am done with.

 

[suziequeweb]

I am a maths tutor, no problem if you don't have it, it just will save me time, as I am not good with sketching on comp! Thanks

 

[fireychariot]

rohan03 how did you solve the problem you had here? I too am struggling with this.

 

[rohan03]

By looking in the textbook again.

 

[fireychariot]

Which method did you end up using though?