Solutions to Selected Questions from Section 1.1.1
September 22, 2010
There are no answers in the back of the text for Section 1.1.1. This document is an attempt to remedy this by providing a half dozen or so solutions to
representative problems from that
Rules for Undetermined Coefficients
We want to find a particular solution, y p , of a nonhomogeneous linear ordinary differential equation
with constant coefficients of the form:
ODE: L ( y ) = an y ( n) + an 1 y ( n 1) +
where
and
1.
2.
+ a1 y + a0 y = g
MATH 817 ASSIGNMENT 1 SOLUTION
(1) Take any a, b G. a2 = b2 = (ab)2 = 1 so [a, b] = a1 b1 ab = (ab)(ab) = 1.
(2) If G = 1 then G has exactly 1 subgroup. If there is any non-identity element of G
which does not generate G then the subgroup generated by thi
MATH 817 ASSIGNMENT 3 SOLUTIONS
(1) Proof by induction. First note that G = G1 , H = H 1 and is surjective so (G1 ) =
H 1 . Suppose n > 1 and suppose inductively that (Gn1 ) = H n1 . Then (Gn ) =
([Gn1 , G]) = [(Gn1 ), (G)] = [H n1 , H] = H n .
(2) Since
MATH 817 ASSIGNMENT 2 SOLUTIONS
(1) (a) G
G so let G act on G by conjugation. The kernel of the action is CG (G ),
and so G/CG (G ) is isomorphic to a subgroup of Aut(G ).
But G is cyclic by assumption, so the only automorphisms consist of raising the
ele
MATH 817 ASSIGNMENT 4 SOLUTIONS
(1) (a) Let
c= A C B
be an object of C . Then dene idc to be idC in C, which is a valid morphism of
C since
A C B
id
id
id
A
C
B
A C B
commutes. If and are morphisms of C such that
A C B
id
id
A
B
A C B
and
A C B
id
id
A
B
MATH 817 ASSIGNMENT 5 SOLUTIONS
(1) Following Isaacs hint let S be the set of all X-subgroups H G for which there is
no nite subset Y G with G = H Y .
Suppose S had a maximal element H. Then H < G since otherwise the empty set
would contradict the conditi
MATH 817 ASSIGNMENT 6 SOLUTIONS
(1) Note rst that eRe is a ring with multiplicative identity e1e = e2 = e. Note also that
for any x eRe, exe = x.
Take any x eJ(R)e. Then x = ere where r J(R). Let I be the ideal
generated by x in eRe. Take any element xs I