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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