MATH 731, FALL 2008
HOMEWORK SET 7
Due Friday, October 31 at noon
A. Suppose A, B , C are subgroups of Gwith A / B ,C / G and suppose BC =G. Prove
that AC is a normal subgroup of G.
B. Suppose Gis an Abelian group that acts transitively on the set X.
(1) Show that Gacts faithfully if and only if g.x =xonly happens when g=e(regardless
of what xis). [Such an action is called free.]
(2) Show that Gacts doubly transitively and faithfully on Xif and only if |G|=|X|= 2
and the non-identity element of Gswaps the two elements of X.
[Hint: Let x∈Xand let g1, g2be elements of Gdifferent from the identity – possibly
g1=g2– and consider the “source pair” x, g2.x and the “target pair” g1.x, x in the
definition of doubly transitive.]
C. (1) Prove that if n≥3, every element of Anis a product of 3 -cycles. [This is true for
all nif we understand the product of zero 3-cycles to be the identity.]
(2) Prove that if n≥5, every element of Anis a product of permutations of the form
(a b)(c d) where a, b, c, d are distinct.
D. Suppose Fis a finite field with qelements (so F=Fq). Explain the following equalities.
(1) |Pn−1(F)|= (qn−1)/(q−1).
(2) |GLn(F)|=Qn−1
i=0 (qn−qi) = qn(n−1)/2·(qn−1)(qn−1−1) · · · (q−1).