Cyclic group

Homework Statement

How do i go about proving that a group is cyclic?

The Attempt at a Solution

dx
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The group, G, is a finte group with cardinality p, a prime integer. How should i start off, if i need to prove it's cyclic?

dx
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Are you familiar with Lagrange's theorem?

yes, i know that the only subgroups of G are itself and the subgroup {e} which consists of the neutral element. This is because the only possibilities of the cardinalities of the subgroups are 1 or p.

dx
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Ok, now pick an element of the group G, say g not equal to 1. What are the possible orders of g?

possible orders of g are 1 or p? Since those are the only numbers that divide the prime number p.

dx
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It cannot be 1 because we assumed g was not equal to the identity. So the order of g must be p, and therefore G = {1 , g, g2, ... , gp-1} which is cyclic.

Ok, thanku very much for the help:)

dx
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No problem.