38 1. Sylow

Note. If |G| = 36, then a Sylow 3 subgroup really can fail to be normal,

but in that case, one can show that a Sylow 2-subgroup of G is normal.

IE.2. Let |G| = pqr, where p q r are primes. Show that nr(G) — 1.

Hint. Otherwise, show by counting elements that a Sylow ^-subgroup must

be normal and consider the factor group, of order pr.

Note. In general, if |G| is a product of distinct primes, then a Sylow sub-

group for the largest prime divisor of \G\ is normal. We prove this theorem

of Burnside when we study transfer theory in Chapter 5.

IE.3. Show that there is no simple group of order 315 =

32«5*7.

Note. This, of course, is also a consequence of the Feit-Thompson odd-order

theorem, which asserts that no group of odd nonprime order can be simple.

1E.4. If \G\ = 144 =

24-32,

show that G is not simple.

1E.5. If \G\ = 336 =

24-3-7,

show that G is not simple.

Hint. If G is simple, compute 717(G) and use Problem 1C.5.

1E.6. If \G\ = 180 =

22-32-5,

show that G is not simple.

1E.7. If |G| = 240 -

24-3-5,

show that G is not simple.

1E.8. If \G\ = 252 =

22-32-7,

show that G is r^ot simple.

I F

Recall that Op(G) is the unique largest normal p-subgroup of the finite

group G, and that it can be found by taking the intersection of all of the

Sylow p-subgroups of G. But do we really need all of them? What is the

smallest collection of Sylow p-subgroups of G with the property that their

intersection is Op(G)l In the case where a Sylow p-subgroup is abelian,

there is a pretty answer, which was found by J. S. Brodkey. (Of course,

since the Sylow p-subgroups of G are conjugate by Sylow C-theorem, they

are isomorphic, and so if any one of them is abelian, they all are.)

1.37. Theorem (Brodkey). Suppose that a Sylow p-subgroup of a finite

group G is abelian. Then there exist £, T e Sylp(G) such that SOT = Op(G).

Of course, every intersection of two Sylow p-subgroups of G contains

Op(G), so what Brodkey's theorem really says is that if the Sylow subgroups

are abelian, then Op(G) is the unique minimal such intersection. In fact,

what is essentially Brodkey's argument establishes something about minimal

intersections of two Sylow p-subgroups even if the Sylow subgroups are not

abelian.