Orders of Elements of Dihedral Group D8 (Correction from earlier post)
Dummit and Foote (D&F) Section 1.2 Dihedral Groups, exercise 1(b) reads as follows:
Compute the order of each of the elements of .
D&F give the following as one presentation of :
= < r,s | = = 1, rs = >
Following this notation I proceeded to compute r, s and sr and (among other elements) was going to determine the order of sr. I proceeded as follows:
Given the set-up mentioned for the square - see attached pages from D&F on the Dihedral Groups - i defined r and s as follows:
So from these permutations I calculated that
= = 1
BUT!! from the relations given by D&F we have = 1 and so we can proceed thus to find
= = 1. ????
Can anyone help?
Re: Orders of Elements of Dihedral Group D8 (Correction from earlier post)
in cycle notation, we have:
r = (1 2 3 4), s = (1 4)(2 3).
so sr = (1 3), a transposition, which is clearly of order 2.
your mistake is in your calculation of (sr)^2 algebraically.
it is NOT the case that (sr)^2 = s^2r^2. typically, that is only true in abelian groups.
what you should have is this:
(sr)^2 = (sr)(sr) = s(rs)r (by associativity)
= s(sr^-1)r (from rs = sr^-1)
= (ss)(r^-1r) = (1)(1) = 1.