By Henry G. Bray
This ebook is ready solvable teams. It focusses on specific sessions of finite solvable teams (supersolvable, M-group, etcetera) giving, in each one case, not less than the fundamental effects approximately that classification. we have now selected to contemplate purely periods which comprise ~. the category of all finite nilpotent teams. This
choice isn't as capricious because it first appears to be like. numerous effects confirmed in bankruptcy five express that ~ is contained in all sessions that are "well-behaved" in a variety of senses (Most striking of those effects is that ~ is contained in all formations which might be in the neighborhood outlined by way of a approach of nonempty formations).
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Extra info for Between Nilpotent and Solvable
Then Z(G) contains an element z of order p with z E (u) and/or z E (y). Induction applied to G / (z) leads readily to the conclusion (u;)(y) = (y)(u;) for each i. Note In general the product of two quasi-central elements is not quasicentral, even when the two elements commute. As an example, let G =(a, b,c: a 3 = b 3 = c 2 = 1, ab= ba, ac =ca, be= cb 2) ~ S 3 X 71. 3• In this (supersolvable) group the elements a and bare both quasi-central. However ab is not since (ab)( c) is not a subgroup of G.
Weak Normality In preceding sections subgroup permutability has been used to generalize the center and hypercenter of finite groups. In this section we develop these ideas further with a concept of "normality" based on the permuting of certain subgroups. This concept leads to a description of supersolvable groups similar to well-known descriptions of nilpotent groups. A knowledge of P. Hall's generalization of the Sylow theorems is crucial for this section. The reader is referred to Appendix A for the necessary concepts, notation, and results.
The permutizer of Hin G is the subgroup PG(H) = (x E G: (x)H = H(x)). Clearly PG(H) ~ NG(H). DEFINITION G is said to satisfy the permutizer condition if H < PG(H) for all H < G. 9 Let G be supersolvable and let M be a maximal subgroup of G. Then PG(M) = G. Proof: Since G is supersolvable, [ G: M] = p, a prime. Hence G contains a p-element x which is not in M. Then from the product formula for subgroups (IHKI =IHI · IKl/IH n Kl), we conclude that (x)M = G so that (x)M = M(x). Therefore PG(M) = G.
Between Nilpotent and Solvable by Henry G. Bray