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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 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 p^k 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) $\not =\{1\}$

See Also

• Frattini subgroup
• p-group
• commutator subgroup

Category:Group theory