2. PERMUTATION GROUPS 3
LEMMA
1.7. Suppose that some p-central element of X of order p lies in
U(X, M,p). Then M controls strong X-fusion in P.
PROOF.
Let x G U(X, M,p) be p-central in X and let P G Sylp(Cx(x)). Then
P G Sylp(X), and indeed P G Sylp(M) since Cx(x) M. Moreover by the Alperin-
Goldschmidt theorem [IQ] 16.1], to show that strong X-fusion in P is controlled by
M it is enough to show that M contains the normalizers Nx(D) of all subgroups
D of P lying in the Alperin-Goldschmidt family X, and so it is enough to show
NX(D) M for any subgroup D of P such that
Op'(Cx(D))
= D. But then
x G Z{P) CP{D) D and so NX(D) M by Lemma 1.4. This completes the
proof.
The following variation is also useful.
LEMMA
1.8. Let x G C and g G X. If Cx(x9) contains an element ofU orUk
for some k, then g G M.
PROOF.
Let Y be an element of U or Uk centralizing x9. Then Y9 lies in
CG(%),
hence in M as x e C. Since Y G hi or Uk, it follows that g~l G M. Thus
g G M, as asserted.
Finally, the conditions defining U are essentially inductive:
LEMMA
1.9. Let H X, Hi H with Hi M, and set H = HjHx and H0 =
H DM (whence Hi H0). Assume that H0 H. If x G H0 with x G U(X, M,p),
then x G U(H,Ho,p).
PROOF.
Let t G H. Then the following conditions are equivalent:
xl
G Ho]
xl G Ho] xl G M (since x and t lie in H and H0 = H D M); t e M (since
x G W(X, M,p)); t £ Ho (since t E H)] i e Ho- In particular if x = 1, then we
conclude that H = Ho and so H = Ho, contrary to assumption. Therefore x has
order p and so x G ^(-£T, Ho,p), as required.
2. Permutation Groups
Throughout this section, Q, is a finite set and X a group acting on Q. We adopt
the subscript notation for fixed point sets:
(2A) For any Y C X, we set ttY = {& G ft | c^ = a for all # G Y}.
Clearly Qy
^Y9 f°r anY
9 X and in particular Oy is invariant under
Nx(Y). If Y" consists of a single element y, we also write Qy for fty. Furthermore,
if X acts faithfully on ft we call X a permutation group on ft.
LEMMA
2.1. Let X 6e transitive on ft witt |ft| 2. Let a,/3 G ft and P G
Sylp(Xap) for some odd prime p. //ftp = {a,/?}; t/ien P G Sylp(X).
PROOF.
Since 7Vx(P) stabilizes ftP = {a,/?}, |iVx(P) : iVx(P)a/3| 2. By
assumption P is a Sylow p-subgroup of Nx(P)a/3 and p is odd, so P is a Sylow
subgroup of NX(P) and therefore of X.
We next prove an elementary but important criterion of Bender for a transitive
group X to be 2-transitive.
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