GT5. Index 2 Theorem and Dihedral Groups

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EDIT: typo at 12:00, it should be "0 less than/equals k less than n", so as to include e and C.

Abstract Algebra: We state and prove the Index Two Theorem for finding normal subgroup and list several examples. These include S3, A4, and the symmetry groups for the regular n-gon, D_2n. We give several presentations of the latter groups and calculate the center.

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Small nitpick: at 12:00 you say: D_2n = {r^k, cr^k | 1 <= k < n} but that misses the identity element!

FreeAsInFreeBeer

@lineopaint You're welcome! I know - abstract algebra and real analysis are tough courses under the best conditions. - Bob

MathDoctorBob

The notation is very close here. G/H is the set of cosets {H, xH}, but G \ H is supposed to mean set complement (take G and throw away the elements of H).

MathDoctorBob

Oh of course. I should have noticed the slash was the other way around. Thanks!

TusksRUs

In your definition of D_2n corresponding to complex numbers, why are we excluding k=0?

Grassmpl

It's a good comment. Worth annotating. - Bob

MathDoctorBob

On the first slide you show, on the right hand side, it says xH = G\H and Hz = G\H

Is this something I don't understand, because I thought G\H was the set containing H and xH, not just the set xH itself?

TusksRUs

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