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.
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