Math 113 Section 5
Homework #2
Fall 2010
Due: Monday, September 13
1.
Determine which of the following sets with
2
to
1
operations form groups. For those
that are groups, prove that they are groups. For those that are not groups, describe
why they are not groups.
(a)
Z
with usual addition
+
(b)
Z
with usual multipication
·
(c)
Z
/n
Z
where
n
is any positive integer, with multiplication
·
(d)
(
Z
/n
Z
)
×
, the set of invertible elements of
Z
/n
Z
, with multiplication
·
(e)
M
n
×
m
, the set of
n
×
m
matrices with entries in
R
, with matrix addition
(f)
M
n
×
m
, the set of
n
×
m
matrices with entries in
R
, with matrix multiplication
(g)
M
n
, the set of (square)
n
×
n
matrices with entries in
R
, with matrix multiplication
(h)
G
=
{
a
+
b
√
2

a, b
∈
Q
} ⊂
R
with multiplication
·
(i)
G
=
{
z
∈
C

z
n
= 1
}
, where
n
∈
Z
>
0
, with multiplication of complex numbers
(j)
Q
where
a
b
is given in lowest terms and
·
is defined by
a
b
·
c
d
=
a
+
c
b
+
d
, reduced to
lowest terms
2.
Let
(
G,
·
)
be a group.
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 Fall '08
 OGUS
 Algebra, Multiplication, Sets, multiplication table

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