- #1
rohan03
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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.
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.
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