User:Ckhenderson

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

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

• ${\displaystyle \Omega _{i}(G)}$ is the set of all elements in G which have order p^k where ${\displaystyle k\leq i}$.
• ${\displaystyle Ag^{i}(G)}$ 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) ${\displaystyle \not =\{1\}}$

• Frattini subgroup
• p-group
• commutator subgroup

Category:Group theory