CDM Semigroups and Groups

ly we know that 〈A 〉 always exists since 〈A 〉 = ⋂ {H ⊆ G | A ⊆ H subgroup } But computationally this is useless: we already have to know all subgroups (containing A) to compute the intersection. Even if G is finite and relatively small, there is no hope to build an algorithm on this characterization. Also note that 〈A 〉 is one of the H’s on the right hand side (impredicative definition). So how do we actually compute 〈A 〉 from A? One Generator 132 If there is only one generator g, and G is finite, then all we get is H = { g | 0 ≤ i < m } where m is the order of g in G. Note that in this case H is Abelian. In fact, H is isomorphic to Zm via i 7→ g. If the order of g is infinite then 〈 g 〉 is isomorphic to Z.