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.