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.