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).
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