School of Mathematics and Statistics MT5824 Topics in Groups ...

15. The purpose of this question is to prove Theorem 6.33; that is, that any two
Sylow bases in a finite soluble group G are conjugate.
Let G be a finite soluble group and let p 1 , p 2 ,...,p k be the distinct prime
factors of G. Let S i be the set of Hall p i-subgroups of G (for i = 1, 2, . . . , k)
and let S be the collection of all Sylow bases of G; that is, the elements of S
are sequences (P i ) such that P i is a Sylow p i -subgroup of G for i = 1, 2, . . . , k
and P i P j = P j P i for all i and j.
(a) Show that
(Q 1 ,Q 2 ,...,Q k ) → Q j , Q j ,...,
Q j
j=1 j=2 j=k
is a bijection from S 1 × S 2 ×···×S k to S .
(b) Now fix a representative Q i in S i . Show that |S i | = |G :N G (Q i )|. Deduce
that |S i | is a power of the prime p i .
(c) Show that G acts on S according to the rule:
(Pi ),g → (P g
i )
for (P i ) ∈ S and g ∈ G. (As usual, P g
i
denotes the conjugate of P i by g.)
(d) Now concentrate on the specific Sylow basis (P i ) constructed from the Q i
as in lectures. [Also compare part (a).] Show that the stabiliser of (P i )
under the above action is the intersection k
i=1 N G(P i ) of the normalisers of
the P i , and that this coincides with the intersection k
j=1 N G(Q j ).
(e) Use part (b) to show that
[Hint: Coprime indices!]
k G :
k
N G (Q j )
= |G :N G (Q j )|.
j=1
j=1
(f) Use part (a) to deduce that G acts transitively on S .
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