Group of Prime Order is Cyclic

In this page, we prove that every group of prime order is cyclic.

Prove that Group of Prime Order is Cyclic

Answer:

Let G be a group of order p where p is a prime number So |G| = p.

As p is a prime number, p>1. That is, |G| > 1. So there is an element a in G (other than identity) of order > 1. That is, $\circ$(a) > 1.

By the application of Lagrange’s theorem, we get that $\circ$(a) divides |G| = p. This implies that

$\circ$(a) = 1 or p.

As $\circ$(a) > 1, we deduce that $\circ$(a) = p. So we have obtained an element a of order p in G, making a is a generator of the group G ⇒ G = <a>.

Therefore, G is a cyclic group. In other words, we have shown that every group of prime order is cyclic.

Questions Answers

Q1: Is a group order 13 cyclic?

Answer:

Since 13 is a prime number and we know that every group of prime order is cyclic, it follows that a group of order 13 is cyclic.

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