# FOM: My BT post again

Fri Dec 12 16:42:04 EST 1997

David Ross wrote:
> As for references, I think the nicest exposition of BT and related
> paradoxes is still Sierpinski's monograph on the Congruence of Sets,
> though he doesn't explicitly identify F2 in his exposition.  Wagon
> probably does in his book on BT, though to be honest I haven't actually

I've read a good part of Wagon's book on the paradox and he does indeed
spend a great deal of time detailing the role of F2 in the decomposition.

He shows that for _any_ paradoxical decomposition using 4 pieces by a group
G acting on some set, then there is a subgroup of G isomorphic to F2
(if I recall correctly).

I do recall that the converse is true (assuming AC):
If G acts on some set X and G has a subgroup iso to F2 then there
is a 'G-paradoxical' decomposition of X.
(It is necessary to assume that G is 'locally commutative' on X, meaning
that if a,b\in G share a fixed point, then they commute.  This is trivially
true if G has no non-trivial fixed points.  It is also true of the group
of isometries of R^n.)

(I find Wagon's book to be a very readable account, despite only having
taken a few graduate math courses.)

--Apollo Hogan

Professional Research Assistant
Dr. Elizabeth Bradley, Dept. of Computer Science