Coset Representatives:
Two Opposing Views

(For some background, see Dharwadker at Wikipedia and
Ashay Dharwadker, the Four Color Theorem, and Usenet Postings.)

by Steven H. Cullinane,
Sept. 7, 2005

From "Common Systems of Coset Representatives,"
by Ashay Dharwadker, Sept. 2005, at
http://www.geocities.com/dharwadker/coset.html --

"Using the axiom of choice, we prove that given any group G and subgroup H, there always exists a common system of coset representatives of the left and right cosets of H in G."

From "Hopf Algebra Extensions and Monoidal Categories" (pdf),
by Peter Schauenburg, at
http://www.mathematik.uni-muenchen.de/~schauen/papers/haemc.pdf --

The image “http://www.log24.com/theory/Dharwadker/CosetReps.jpg” cannot be displayed, because it contains errors.

Update of Sept. 11, 2005:

Dharwadker has changed his claim. He now says,

"Using the axiom of choice, we prove that given any group G and a finite subgroup H, there always exists a common system of coset representatives of the left and right cosets of H in G."

This new claim avoids the difficulty described by Schauenburg above, since H is now assumed to be finite. It also meets the following conditions stated by Fred Galvin (sci.math, Feb. 20, 2003):

"... if H is a finite subgroup of a (finite or infinite) group G, then there is a common transversal for the system of left cosets and the system of right cosets. This is still true for an infinite subgroup of finite index, but it breaks down for infinite subgroups of infinite index.

See L. Mirsky, Transversal Theory, Academic Press, 1971, ISBN 0-12-498550-5."