# Prime order groups are cyclic

Let $$G$$ be a group whose or­der is equal to $$p$$, a prime num­ber. Then $$G$$ is iso­mor­phic to the cyclic group $$C_p$$ of or­der $$p$$.

# Proof

Pick any non-iden­tity el­e­ment $$g$$ of the group.

By La­grange’s the­o­rem, the sub­group gen­er­ated by $$g$$ has size $$1$$ or $$p$$ (since $$p$$ was prime). But it can’t be $$1$$ be­cause the only sub­group of size $$1$$ is the triv­ial sub­group.

Hence the sub­group must be the en­tire group.

Parents:

• Group

The alge­braic struc­ture that cap­tures sym­me­try, re­la­tion­ships be­tween trans­for­ma­tions, and part of what mul­ti­pli­ca­tion and ad­di­tion have in com­mon.