User:Ckhenderson

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 $$i\in\mathbb{N}, \Omega_i(G) = < \{g : g^{p^i} = 1 \} > $$. The Agemo subgroups are the class of subgroups, $$ Ag^i(G) = < \{ g^{p^i} : g \in G \} >$$.

Both the Agemo and Omega subgroups play important roles in many proofs many properties about p-groups,

Some Facts

 * $$\Omega_i(G)$$ is the set of all elements in G which have order pk where $$ k \leq i $$.
 * $$Ag^i(G)$$ 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) $$ \not= \{1\} $$