Equivalence of definitions of group of prime power order

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This article gives a proof/explanation of the equivalence of multiple definitions for the term group of prime power order
View a complete list of pages giving proofs of equivalence of definitions

Statement

The following are equivalent for a finite group and a prime p:

  1. The order of the group is a power of p.
  2. The order of every element in the group is a power of p.

Facts used

  1. Order of element divides order of group
  2. Cauchy's theorem: This states that if q is a prime dividing the order of a finite group, there exists an element in the finite group of order q.

Proof

(1) implies (2)

This follows directly from fact (1).

(2) implies (1)

Suppose P is a finite group satisfying (2). Then, by fact (2), if q is any prime other than p dividing the order of P, P has anelement of order q, contradicting (2). Thus, p is the only prime dividing the order of P, proving (1).