Problems IB 13

(b) Show that every Sylow p-subgroup of K has the form 0(H), where

H is some Sylow p-subgroup of G.

(c) Show that |Sylp(G)| \Sy\p(K)\ for every prime p.

N o t e . If the set TT contains more than one prime number, then a Hall TT-

subgroup can fail to exist. But a theorem of P. Hall, after whom these

subgroups are named, asserts that in the case where G is solvable, Hall TT-

subgroups always do exist. (See Chapter 3, Section C.) We mention also

that Part (b) of this problem would not remain true if "Sylow p-subgroup"

were replaced by "Hall 7r-subgroup".

I B . 6 . Let G be a finite group, and let K C G be a subgroup. Suppose that

H C G is a Hall 7r-subgroup, where TT is some set of primes. Show that if

HK is a subgroup, then H n K is a Hall 7r-subgroup of K.

Note . In particular, K has a Hall 7r-subgroup if either H or K is normal in

G since in that case, HK is guaranteed to be a subgroup.

I B . 7 . Let G be a finite group, and let TT be any set of primes.

(a) Show that G has a (necessarily unique) normal 7r-subgroup T V such

that N I) M whenever M G is a 7r-subgroup.

(b) Show that the subgroup N of Part (a) is contained in every Hall

7r-subgroup of G.

(c) Assuming that G has a Hall 7r-subgroup, show that N is exactly

the intersection of all of the Hall 7r-subgroups of G.

Note . The subgroup N of this problem is denoted 07T(G). Because of the

uniqueness in (b), it follows that this subgroup is characteristic in G. Finally,

we note that if p is a prime number, then, of course, 0{py(G) — Op(G).

I B . 8 . Let G be a finite group, and let TT be any set of primes.

(a) Show that G has a (necessarily unique) normal subgroup N such

that G/N is a 7r-group and M D N whenever M G and G/M is

a 7r-group.

(b) Show that the subgroup N of Part (a) is generated by the set of all

elements of G that have order not divisible by any prime in TT.

Note . The characteristic subgroup N of this problem is denoted

Oir(G).

Also, we recall that the subgroup generated by a subset of G is the (unique)

smallest subgroup that contains that set.