MATH 731, FALL 2008
HOMEWORK SET 6
Due Friday, October 24 at noon
A. Suppose that Gis a group with |G|=pnfor some prime number pand nonnegative
integer n. Show that Gis polycyclic. (You may use facts you know about such groups
from 631–632.)
B. Recall that the derived series of a group Gis defined by G(0) =G,G(k+1) = [G(k), G(k)].
For example, G(1) = [G, G] is the commutator subgroup. Recall also that Gis solvable
iff some G(n)={e}.
(1) If φ:G→His a group homomorphism, show that for any k, we have
−→
φ(G(k))⊆H(k).
(2) Use (1), with H=Gand appropriate choices of φ, to show that each G(k)is a
normal subgroup of G.
(3) If φ:G→His onto, show that for any k, we have −→
φ(G(k)) = H(k).
(4) By making appropriate choices for φ, G, H in (1) and (3), show that any subgroup of
a solvable group is solvable and and any quotient group of a solvable group is solvable.
C. Let Fbe a field and let Jbe an n×nmatrix over F. Define
G(J) = {A∈GLn(F)|AtJA =J}.
(1) Show that G(J) is a subgroup of GLn(F).
(2) Show that if Pis a nonsingular n×nmatrix, then G(J)∼
=G(P J P t).