176 Epilogue for #1, #2 • • • ft £ G, as well as the set x = {(9u92,--,9P) € Gp | 9x92'"9V = e } , where e again denotes the group identity. Prove the following: • \X\ = \G\P-\ • The mapping (p maps the set X into itself. • If the identity (pk(x) = x holds for an element x G Gp and an integer k not divisible by p, then all the coordinates of x are equal. • Every orbit { x, (f(x), ^2(:r),... }, where x £ Gp, consists of either one element or p elements. • The number of one-element orbits in X is divisible by p. Assuming that there exists an element x G X with a one- element orbit, conclude that there exists at least one other ele- ment with a one-element orbit, and thereby prove the existence of an element of G of order p (Cauchy's theorem).11 Cauchy's theorem is usually formulated in a more general form, which is named after Ludwig Sylow (1832—1918). The Sylow theorems make assertions about subgroups of a group of order the power of a prime.

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