Suppose a prime \(p\) divides the order of a finite group \(G\). Prove that there exists an element \(g\in G\) of order \(p\).
Solution
Consider the set \(X = \{ ( g _ { 1 } , \dotsc , g _ { p } ) \in G ^ { p } \mid g _ { 1 } \cdot \cdot \cdot g _ { p } = e \}\) . It is acted upon by the cyclic group \(\mathbf { Z } / p \mathbf { Z }\) with \({ \bf 1 } \in { \bf Z } / p { \bf Z }\) acting as the cyclic shift
\[ ( g _ { 1 } , \dotsc , g _ { p } ) \longmapsto ( g _ { p } , g _ { 1 } , \dotsc , g _ { p - 1 } ) . \]
A fixed point of this action is a constant p-tuple \(( g , \ldots , g )\) such that \(g ^ { p } = e\) . The number of fixed points is not zero, since \(( e , \ldots , e )\) is a fixed point, and is congruent modulo \(p\) to
\[ | X | = | G | ^ { p - 1 } , \]
i.e., it is divisible by \(p ,\) since \(p > 1\) . It follows that there is an element \(g \neq e\) with \(g ^ { p } = e\)