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Omega and agemo subgroup
In mathematics, or more specifically group theory, the omega and agemo subgroups described the socalled "power structure" of a finite pgroup. They were introduced in (Hall 1933) where they were used to describe a class of finite pgroups whose structure was sufficiently similar to that of finite abelian pgroups, the socalled, regular pgroups. The relationship between power and commutator structure forms a central theme in the modern study of pgroups, as exemplified in the work on uniformly powerful pgroups.
The word "agemo" is just "omega" spelled backwards, and the agemo subgroup is denoted by an upsidedown omega.
Definition
The omega subgroups are the series of subgroups of a finite pgroup, G, indexed by the natural numbers:
\( \Omega_i(G) = \langle \{g : g^{p^i} = 1 \} \rangle. \)
The agemo subgroups are the series of subgroups:
\( \mho^i(G) = \langle \{ g^{p^i} : g \in G \} \rangle. \)
When i = 1 and p is odd, then i is normally omitted from the definition. When p is even, an omitted i may mean either i = 1 or i = 2 depending on local convention. In this article, we use the convention that an omitted i always indicates i = 1.
Examples
The dihedral group of order 8, G, satisfies: ℧(G) = Z(G) = [ G, G ] = Φ(G) = Soc(G) is the unique normal subgroup of order 2, typically realized as the subgroup containing the identity and a 180° rotation. However Ω(G) = G is the entire group, since G is generated by reflections. This shows that Ω(G) need not be the set of elements of order p.
The quaternion group of order 8, H, satisfies Ω(H) = ℧(H) = Z(H) = [ H, H ] = Φ(H) = Soc(H) is the unique subgroup of order 2, normally realized as the subgroup containing only 1 and −1.
The Sylow psubgroup, P, of the symmetric group on p^{2} points is the wreath product of two cyclic groups of prime order. When p = 2, this is just the dihedral group of order 8. It too satisfies Ω(P) = P. Again ℧(P) = Z(P) = Soc(P) is cyclic of order p, but [ P, P ] = Φ(G) is elementary abelian of order p^{p}^{−1}.
The semidirect product of a cyclic group of order 4 acting nontrivially on a cyclic group of order 4,
\( K = \langle a,b : a^4 = b^4 = 1, ba=ab^3 \rangle, \)
has ℧(K) elementary abelian of order 4, but the set of squares is simply { 1, aa, bb }. Here the element aabb of ℧(K) is not a square, showing that ℧ is not simply the set of squares.
Properties
In this section, let G be a finite pgroup of order G = pn and exponent exp(G) = pk have a number of useful properties.
General properties
 Both Ω_{i}(G) and ℧^{i}(G) are characteristic subgroups of G for all natural numbers, i.
 The omega and agemo subgroups form two normal series:

 G = ℧^{0}(G) ≥ ℧^{1}(G) ≥ ℧^{2}(G) ≥ ... ≥ ℧^{k−2}(G) ≥ ℧^{k−1}(G) > ℧^{k}(G) = 1
 G = Ω_{k}(G) ≥ Ω_{k−1}(G) ≥ Ω_{k−2}(G) ≥ ... ≥ Ω_{2}(G) ≥ Ω_{1}(G) > Ω_{0}(G) = 1
 and the series are loosely intertwined: For all i between 1 and k:
 ℧^{i}(G) ≤ Ω_{k−i}(G), but
 ℧^{i−1}(G) is not contained in Ω_{k−i}(G).
Behavior under quotients and subgroups
If H ≤ G is a subgroup of G and N ⊲ G is a normal subgroup of G, then:
 ℧^{i}(H) ≤ H ∩ ℧^{i}(G)
 Ω_{i}(H) = H ∩ Ω_{i}(G)
 ℧^{i}(N) ⊲ G
 Ω_{i}(N) ⊲ G
 ℧^{i}(G/N) = ℧^{i}(G)N/N
 Ω_{i}(G/N) ≥ Ω_{i}(G)N/N
Relation to other important subgroups
 Soc(G) = Ω(Z(G)), the subgroup consisting of central elements of order p is the socle, Soc(G), of G
 Φ(G) = ℧(G)[G,G], the subgroup generated by all pth powers and commutators is the Frattini subgroup, Φ(G), of G.
Relations in special classes of groups
In an abelian pgroup, or more generally in a regular pgroup:
 ℧^{i}(G)⋅Ω_{i}(G) = G
 [℧^{i}(G):℧^{i+1}(G)] = [Ω_{i}(G):Ω_{i+1}(G)],
where H is the order of H and [H:K] = H/K denotes the index of the subgroups K ≤ H.
Applications
The first application of the omega and agemo subgroups was to draw out the analogy of regular pgroups with abelian pgroups in (Hall 1933).
Groups in which Ω(G) ≤ Z(G) were studied by John G. Thompson and have seen several more recent applications.
The dual notion, groups with [G,G] ≤ ℧(G) are called powerful pgroups and were introduced by Avinoam Mann. These groups were critical for the proof of the coclass conjectures which introduced an important way to understand the structure and classification of finite pgroups.
References
Dixon, J. D.; du Sautoy, M. P. F.; Mann, A.; Segal, D. (1991), Analytic propgroups, Cambridge University Press, ISBN 0521395801, MR 1152800
Hall, Philip (1933), "A contribution to the theory of groups of primepower order", Proceedings of the London Mathematical Society 36: 29–95, doi:10.1112/plms/s236.1.29
LeedhamGreen, C. R.; McKay, Susan (2002), The structure of groups of prime power order, London Mathematical Society Monographs. New Series 27, Oxford University Press, ISBN 9780198535485, MR 1918951
McKay, Susan (2000), Finite pgroups, Queen Mary Maths Notes 18, University of London, ISBN 9780902480179, MR 1802994
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