Symmetries and Transformation Groups of Equilateral Triangle & Icosahedron

[koolmodee]

How many symmetries (and what symmetries) and how many elements do the transformation groups of the equilateral triangle and the icosahedron have?

thanks

 

[CompuChip]

|D₃| and |A₅ x Z₂|, respectively.

How about you start by writing down those of the equilateral triangle. There aren't that many so you can easily list them.

 

[koolmodee]

3 reflections and 3 rotations, right? But what does |D3| mean?
And what is |A5 x Z2|?

Where can I look up the symmetries for other geometric objects, like for example the other platonic solids?

And what about the number of elements, how do i find out about those?

 

[adriank]

If you've studied groups much, you should recognize what each of those groups are. D₃ is the dihedral group of order 6, A₅ × Z₂ is the direct product of A₅, the alternating group of degree 5, which has 5!/2 = 60 elements, and Z₂, the cyclic group of order 2.

Thus, the equilateral triangle has 6 symmetries, and the icosahedron has 120.