Cyclic subgroup

All prime-ordered groups are cyclic

If a group 𝐺 has prime order 𝑝 =|𝐺|, then it must be cyclic, group i.e. isomorphic to ℤ+𝑝.

Proof

Let 𝐺 be a group of prime order 𝑝. By Lagrange’s theorem, 𝐺 only has subgroups of order 1 and 𝑝. Since 𝑝 >1, there exists 𝑎 ∈𝐺 such that 𝑎 ≠𝑒. Then |𝑎| =𝑝 and therefore ⟨𝑎⟩ =𝐺.

Clearly ℤ+𝑝 is a simple group.

See also

• p-group

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