By THE MATHEMATICAL ASSOCIATION OF AMERICA
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Additional resources for American Mathematical Monthly, volume 105, number 2, February 1998
Tnr+1/G must be nonorientable. Now, the Euler characteristics of any covering space and its orbit space are related by the formula IGI . X(X/G) = X(X). Thus, x(Tnr+1/G) = X(T nr +1) IGI -2nr = - - = -r. 2, if G acts freely, orientation-reversing on T" r+ 1 with IG I = 2n, then there exists an associated exact sequence 1 ~ 'TT'lT nr +1 ~ 'TT'lU r+2 ~ G ~ 1. 146 A COMPUTER SEARCH FOR FREE ACTIONS ON SURFACES [February On the other hand, suppose we begin with an epimorphism a: 7T 1U r +2 ~ G, where G is a finite group of even order.
The price he had to pay was that multiplication in the real quaternions is non-commutative. Mter the construction of the real quaternions, the search for other examples of division algebras was rather slow. , those whose centre contained IR and that are finite-dimensional over IR) are the real numbers, the complex numbers, and the quaternions; this was also proved independently by C. S. Peirce [Pel Some division rings that are not finite-dimensional over their centres were constructed for use in geometry; there is such an example in Hilbert's Foundations of Geometry [Hi, Sect.
He finally constructed the real quaternions, a division algebra that is four-dimensional over its centre, the real numbers. The price he had to pay was that multiplication in the real quaternions is non-commutative. Mter the construction of the real quaternions, the search for other examples of division algebras was rather slow. , those whose centre contained IR and that are finite-dimensional over IR) are the real numbers, the complex numbers, and the quaternions; this was also proved independently by C.