Groups of Prime Order

Abstract Algebra · Axiom Academy

A beautiful and complete classification: every group of prime order is cyclic with no proper subgroups, all isomorphic to ℤ_p. Examples include groups of orders 2, 3, 5, 7, 11, and so on. The canonical example is the additive group of integers modulo p, denoted ℤ_p. 2. Lagrange's Theorem Constrains Everything When |G| = p is prime, the only divisors of p are 1 and p itself. This severely restricts what subgroups can exist! If |H| = 1, then H = e (the trivial subgroup) If |H| = p, then H = G (the whole group) 3. Every Non-Identity Element Generates G Take any element g ≠ e in our group G of order p. Consider the cyclic subgroup ⟨g⟩ generated by g. ⟨g⟩ is a subgroup of G, so by Lagrange's theorem, |⟨g⟩| divides p Since g ≠ e, we have |⟨g⟩| ≥ 2 The only divisors of p are 1 and p But |G| = p as well, so ⟨g⟩ = G 4. The Complete Classification Every non-identity element is a generator Why are they all isomorphic to ℤ_p? Pick any generator g ∈ G. Then G = e, g, g², ..., g^(p-1) , and the map φ: ℤ_p → G defined by φ(k) = g^k is an isomorphism. This stands in stark contrast to composite orders. For example, there are two non-isomorphic groups of order 4: ℤ_4 (cyclic) and ℤ_2 × ℤ_2 (Klein four-group). But for prime orders, there's no such ambiguity.

This is the written version of the interactive lesson above. See the full Abstract Algebra course.