By D. J. H. Garling

ISBN-10: 1107096383

ISBN-13: 9781107096387

ISBN-10: 1107422191

ISBN-13: 9781107422193

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N for r 1, which is even smaller than r. If rt'7~'o, we find q2 and r2 such that and if r2 =0, (m,n)=r 1• If r2 *0, then we continue the process, obtaining a sequence of remainders r>r 1 >r2 >r3 > ... ,where each r;;;il>O. By the wellordering principle, such a decreasing sequence of nonnegative integers cannot go on forever, so some r; must eventually be 0. If so, then (m,n) = (n,r 1) = (r 1,r2 ) = · · · = (r;_ 1,r;) = (r;_ pO) = r;_ 1 • Thus (m,n) is the last nonzero remainder arising from our repeated divisions.

In the above theorem and, in fact, it is not hard to show that it is really sufficient to check either one of them. 5 Let ( G, *) be a group and let x, yE G. Suppose that either x *y = e or y * x = e. Then y is x- 1• Suppose that x *y =e. We wish to solve this equation for y, so let's multiply both sides by x-I: PROOF. Section 3. Fundamental Theorems about Groups 29 Thus (x- 1•x)•y=x- 1, e•y=x-I, y=x-1. 6 (Cancellation laws) Let (G,•) be a group and let x, y, zEG. Then: i) if x•y=x•z, theny=z; and ii) ify•x=Hx, theny=z.

### Clifford algebras : an introduction by D. J. H. Garling

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