- #1
Mr Davis 97
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Homework Statement
Show that ##M = \langle r,s \mid r^m = e, s^n = e, srs^{-1} = r^j \rangle##, where ##j## is a natural number satisfying ##\operatorname{gcd}(j,m) = 1## and ##j^n \equiv 1 \pmod{m}##, has ##mn## elements,
Homework Equations
The Attempt at a Solution
I'm not sure how to start to show this rigorously, but I do have some ideas. If we can show that every element can be written in the form ##r^as^b## then clearly we would have ##m## choices for ##a## and ##n## choices for ##b##, and hence by multiplying there would be ##mn## elements. But I'm not sure how to show that every product can be written in this form. I thought that maybe looking at the dihedral group and doing something analogous would help me, but all of the proofs that the dihedral group has ##2n## elements that I've seen use geomretical reasoning, which I don't seem to be able to do here.
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