18 1. Sylow Theory

Problems 1C

IC.l. Let P G Sylp(3), and suppose that NG(P) C f f C G , where H is a

subgroup. Prove that H =

NG(H).

Note. This generalizes Problem IB.3.

1C.2. Let H C G, where G is a finite group.

(a) If P G Sy\p(H), prove that P = H H S for some member S G

Sylp(G).

(b) Show that np(H) np(G) for all primes p.

1C.3. Let G be a finite group, and let X be the subset of G consisting of

all elements whose order is a power of p, where p is some fixed prime.

(a) Show that X = IJSylp(G).

(b) Show that if p divides |G|, then |X| is divisible by p.

Hint. For (b), let a Sylow p-subgroup act on X.

1C.4. Let \G\ = 120 =

23-3-5.

Show that G has a subgroup of index 3 or a

subgroup of index 5 (or both).

Hint. Analyze separately the four possibilities,for 77,2(6?).

1C.5. Let P G Sylp(G), where G = Ap+i, the alternating group on p + 1

symbols. Show that |N

G

(P)| = p(p - l)/2.

Hint. Count the elements of order p in G.

1C.6. Let G = HK, where H and K are subgroups, and fix a prime p.

(a) Show that there exists P e Sy\p(G) such that P HH G Sylp(H)

trndPHK eSylp(K).

(b) If P is as in (a), show that P = (P n H){P D K).

Hint. For (a), first choose Q G Sylp(G) and g G G such that Q n H G

Sylp(H) and Q9 n K G Sy\,(K). Write g = hk, with /i G if and fe G K.

1C.7. Let G be a finite group in which every maximal subgroup has prime

index, and let p be the largest prime divisor of |G|. Show that a Sylow

p-subgroup of G is normal.

Hint. Otherwise, let M be a maximal subgroup of G containing N^(P),

where P G Sylp(G). Compare np(M) and np(G).