Frieze Patterns

[Guest GA]

Hi,

 

I have the following problem:

 

Consider the infinite strip pattern: ... TTTTTT ...

 

Prove that every element of the symmetry group of the above pattern has the form t^n or (t^n)r where t is a suitable translation and r is a suitable reflection. Is this group Abelian?

 

I can see clearly why every element of the symmetry group has either of those forms, basically just because every isometry can be proven to be either a translation, a reflection, a rotation or a glide reflection and clearly neither a glide reflection nor a rotation will yield symmetry. Then clearly we take t to be the translation 'one unit' to the right or left and then r as a reflection through the vertical line passing through any one of the T's, but how on earth do I go about proving this rigorously?

 

Then it is quite obvious that this group is not Abelian because a translation followed by a reflection is not necessarily the same transformation as that reflection followed by that translation (should be easy enough to prove with a counter-example).

 

Any help on proving this properly?

 

Thanks in advance,

 

Gary.