Soluble and Nilpotent Groups
The calculus of commutators is fundamental to this study; in this section we will establish our notation and some basic properties.
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Soluble and Nilpotent Groups
2.1 Conjugates and Commutators. Some Generalized Soluble and Nilpotent Groups The calculus of commutators is fundamental to this study; in this section we will establish our notation and some basic properties. Let x, y, z, ... be elements of a group. Then, of course, the conjugate ot x by y is and the commutator ot x with y is [x, y] = r1y-1xy = r1x Y.
More genE'rally, simple commutators of weight n tively by the rules [Xl' ... , X ni
d
= [[Xl' "',
X n ], xn-CJ]
+ 1 are defined induc-
and [xd =
Xl'
As is well-known, the identities
LX,
yz] = [x, z] [x, y]"
(1)
[xy, z] = [X, z]y [y, z]
(2)
and (3)
are valid in any group. Verification of (1) and (2) is immE'diate, while (3) is most readily proved by setting 21 = xzrIyx, v = yxy-Izy and w = zyz-Ixz and observing that u-1v = [x, y-I, zy, v-1w = [y, Z-I, x]' and w-Iu = [z, rl, yY: the truth of (3) is now evident. It is important to be able to form conjugates and commutators of subsets as WE'll as elements of a group and these are defined in the following way. Let X, Y, Z, ... be non-empty subsets of a group. The conjugate of X by Y or, as it is usually called, the normal closure ot X in Y, is the subgroup
Xl'
=
G' = Y2(G). This result is usually referred to as Grun's Lemma (Grun [lJ, ยง 2, Satz 4). However it is interesting that apart from this there is very little relation between the lengths of the upper and lower central series, as has been demonstrated by Meldrum [lJ. The last term of the upper central series of G is the hypercentre,
'(G).
2.1 Conjugates and Commutators
49
Most of these ideas have already appeared in Chapter 1 in the more general setting of subgroup theoretical properties. A group G is said to be nilpotent if it has a central series of finite length. It follows easily from the definitions that a group G is nilpotent if and only if Yc+t(G) = 1 for some integer c? 0, and that Yc+1(G) = 1 if and only if Cc(G) = G. The least integer c such that YC+1(G) = lor, equivalently, such that CAG) = G, is called the nilpotent class of G. A fundamental fact about 91, the class of nilpotent groups, is that it is No-closed. More precisely there is the well-known result Theorem 2.18 (Fitting [lJ, p. 100). Let H = .
Hence yX E I and in a similar way X Y E I. But Xl"
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