Dummit and Foote in Section 1.2 Dihedral Groups give the following presentation of
= <r,s | = = 1, rs = s >
Exercise 7 at the end of D&F Section 1.2 is as follows:
Show that <a,b | = = = 1>
gives a presentation for in terms of the two generators a = s and b = sr. [Show that the relations for r and s follow from the relations for a and b and conversely that the relations for a and b follow from those for r and s]
I was able (to my satisfaction anyway) to show all relations followed except for assuming the relations for r and s and trying to show that = 1
In this case I, of course, started with = = 1, rs = s
and then put s = a and r = b = b
Then = = = 1
But I can get no further ... can anyone help?
A further question is this if
(1) we assume the presentation in terms of r and s does as D&F assert generates the whole group
(2) another presentation such as the one above based on a and b is such that the relations involved can be derived from the first
- then we can conclude that the new presentation can be used to derive the whole group? Is that right?