1. NOTATION 3

Section 3. The symbol X * Y refers to a central product of subgroups X and Y, i.e.

a group which is the product XY of subgroups satisfying [X, Y] = 1. Such a group

is a factor group of X x Y in which central subgroups Zx and Zy of X and Y are

identified, and the isomorphism type of X * Y is determined by these subgroups.

Unless otherwise stated, when we use the symbol l * F w e assume that Zx and Zy

are as large as possible among central subgroups of X and Y which are isomorphic.

For any prime p or set n of primes, Op(X), Op/(X), Op/p(X),

Op(X), Op

(X),

O^X) , (^(X) , etc. have their usual meanings as does m

p

(X), the p-rank of X.

Thus 07T(X) is the subgroup generated by all Tr'-elements of X. The set of all

elements of X of order p is TP(X). A p-local subgroup of X is a subgroup Nx{P)

where P is a nontrivial p-subgroup of X. The important subgroups Ln'(X) and

Lvi (X) will be introduced in Section 4.

Multiple commutators will always be implicitly left-associated, so that [x, y, z]

means [ [x, y],z]. HA and B are subsets of X, then [A, B] is the subgroup gener-

ated by all commutators [a, b] with a E A and b E B. Likewise [A, B, C] means

[[A,i?],C]. We write [A, b] in place of [A, {6}]. We mention the important prop-

erties that if A and B are subgroups of X, then [A,B] (A,B); moreover A

normalizes B if and only if [A,B] B.

We introduce the following additional notation for any prime p, which will

likewise be used throughout. Epr is an elementary abelian p-group of order

pr

(thus Ep — Zp and Ep2 — Zp x Zp in the usual notation).

£P(X) = {A X | A ^ Epr, r 2}.

££(X) = {A G 8p(X) | mp(A) k}.

£S(X) = {A G

8p(X)

| mp(A) = mp(X)}.

Tp{X) - {x G XP(X) | mp(Cx(x)) - m

p

(X)}.

A group isomorphic to £4 is a four-group. A maximal subgroup of an elementary

abelian group A is a hyperplane of A.

Also, as usual, we feel free to drop the subscript or superscript p from the

above symbols and mp(X) when the context makes clear what prime p is to be—

for example, if X is a p-group.

On the other hand, we do not drop the symbol p from the terms Op(X), Ov (X),

etc., except for the single case of 02'(X), which is uniformly written O(X).

If X is a p-group, then again as usual, we denote by Qi(X) and

15Z(X),

i 1,

the subgroups of X generated, respectively, by all elements of X of order

p%

and

by the

pl

powers of all elements of X. Also, D2n, n 2, SD2n, n 4, and Q2™,

n 3, denote a dihedral, semidihedral and (generalized) quaternion group of order

2n,

2n

and

2n,

respectively. In particular, D4 is a four-group.

If P is a p-subgroup of X of rank at least £;, where k is a positive integer, then

we define

TPik(X) = (NX(Q) I Q P and mp(Q) k).

In connection with permutation groups, we mention that the set of right cosets

of a subgroup Y in a group X is Y\X; a transversal to Y in X is a set of repre-

sentatives of these cosets. The symmetric and alternating groups on a set ft or on

n letters are EQ, An, E

n

and An. (Other simple and almost simple groups arise

very occasionally; see [Ir,§l] for our notation.) If X acts on a set ft, the action is

semiregular if and only if the point stabilizer Xa is the identity for each a G ft; the

action is regular if and only if it is semiregular and transitive. Furthermore, if X