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 :(
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