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In Mathematics, or more specifically Group Theory, the Omega class of subgroups are the series of subgroups of a finite p-group, G, indexed by the natural numbers, where given . The Agemo subgroups are the class of subgroups, .
Both the Agemo and Omega subgroups play important roles in many proofs many properties about p-groups,
- is the set of all elements in G which have order pk where .
- is the smallest group containing all elements of order pi.
- If G is a finite p-group, then Φ(G) = [G,G] , where [G,G] is the commutator subgroup of G and Φ(G) is the Frattini subgroup of G.
- It follows from Cauchy's theorem that if G is a finite group and p is a prime number which divides the order (group theory) of G, then Ω1(G)