Refined Central Series
In the study of p-groups, it is often important to use longer central series. An important class of such central series are the exponent-p central series; that is, a central series whose quotients are elementary abelian groups, or what is the same, have exponent p. There is a unique most quickly descending such series, the lower exponent-p central series λ defined by:
- λ1(G) = G, and
- λn + 1(G) = (λn(G))p
The second term, λ2(G), is equal to Gp = Φ(G), the Frattini subgroup. The lower exponent-p central series is sometimes simply called the p-central series.
There is a unique most quickly ascending such series, the upper exponent-p central series S defined by:
- S0(G) = 1
- Sn+1(G)/Sn(G) = Ω(Z(G/Sn(G)))
where Ω(Z(H)) denotes the subgroup generated by (and equal to) the set of central elements of H of order dividing p. The first term, S1(G), is the subgroup generated by the minimal normal subgroups and so is equal to the socle of G. For this reason the upper exponent-p central series is sometimes known as the socle series or even the Loewy series, though the latter is usually used to indicate a descending series.
Sometimes other refinements of the central series are useful, such as the Jennings series κ defined by:
- κ1(G) = G, and
- κn + 1(G) = (κi(G))p, where i is the smallest integer larger than or equal to n/p.
The Jennings series is named after S. A. Jennings who used the series to describe the Loewy series of the modular group ring of a p-group.
Read more about this topic: Central Series
Famous quotes containing the words refined, central and/or series:
“A refined soul is distressed to know that someone owes it thanks; a crude soul, to know that it owes someone thanks.”
—Friedrich Nietzsche (18441900)
“Parental attitudes have greater correlation with pupil achievement than material home circumstances or variations in school and classroom organization, instructional materials, and particular teaching practices.”
—Children and Their Primary Schools, vol. 1, ch. 3, Central Advisory Council for Education, London (1967)
“Depression moods lead, almost invariably, to accidents. But, when they occur, our mood changes again, since the accident shows we can draw the world in our wake, and that we still retain some degree of power even when our spirits are low. A series of accidents creates a positively light-hearted state, out of consideration for this strange power.”
—Jean Baudrillard (b. 1929)