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
O^X) , (^(X) , etc. have their usual meanings as does m
(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
(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
| mp(A) = mp(X)}.
Tp{X) - {x G XP(X) | mp(Cx(x)) - m
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
i 1,
the subgroups of X generated, respectively, by all elements of X of order
by the
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
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
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
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