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.