MATH 74 HOMEWORK 7
“If you don’t like your analyst, see your local algebraist!”
–Gert Almkvist
Due Wednesday April 8th at 3:10pm. Throughout, let
G
be a group. Many of the following problems are
from (or adapted from) Herestein’s
Topics in Algebra
.
(1) In the following, decide and supply reasoning as to whether the given systems are groups. If they
aren’t point out an axiom that fails.
(a) the set of integers with
a
·
b
:=
a

b
.
(b) the set of positive integers with
a
·
b
:=
ab
(the usual product of integers).
(c) the set of all rational numbers with odd denominators, with the usual additional of rational
numbers.
(2) Given
a, b
∈
G
, prove that the equations
a
·
x
=
b
and
y
·
a
=
b
have unique solutions for
x, y
∈
G
.
In particular, show we have cancellation laws:
a
·
u
=
a
·
w
=
⇒
u
=
w,
u
·
a
=
w
·
a
=
⇒
u
=
w.
(3) For an element of a group, let
a
n
denote the
n
fold multiplication of
a
with itself. If
G
is a group
with (
a
·
b
)
2
=
a
2
·
b
2
, prove that
G
is abelian.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 DANBERWICK
 Algebra, Division, Ring, nonzero real numbers

Click to edit the document details