Homework SourseDefine a function f from Z1212 to U5multiplication 5 by fn2n
Define a function f from (Z12,+12) to (U(5),multiplication 5) by f(n)=2^n mod 5. Determine if f is a homomorphism or not.
**(The set of units, denoted by U(n), is the set of all positive integers which are relatively prime to n)
Solution
Ans-
Homework 7 Solutions
March 17, 2012
1 Chapter 9, Problem 10 (graded)
Let
G
b e a cyclic group. That is,
G
=
h
a
i
for some
a
2
G
. Then given any
g
2
G
,
g
=
a
n
for some integer
n
.
Let
H
b e any normal subgroup of
G
(actually, since
G
is cyclic, it is also
Ab elian, so all subgroups of
G
are normal), and consider the factor group
G=H
=
f
gH
:
g
2
G
g
.
G=H
is the group whose elements are left cosets of
H
. Let
gH
b e any element of
G=H
. Since
g
=
a
n
for some integer
n
, we have
gH
=
a
n
H:
Next, by denition of multiplication in a factor group,
gH
=
a
n
H
= (
aH
)
n
:
Therefore, if
gH
is any element of
G=H
, then
gH
= (
aH
)
n
for some integer
n
.
This implies that
G=H
=
h
aH
i
. That is,
G=H
is a cyclic group generated by
the element
aH
.
2 Chapter 9, Problem 16 (graded)
Before presenting the solution, let me talk ab out computing order in a factor
group
G=H
. Supp ose
gH
is an element of
G=H
(so
g
2
G
) and I want to
compute its order as an element of
G=H
. In other words, I want to nd an
integer
n
such that
(
gH
)
n
=
eH
=
H
and if
1
m < n
,
(
gH
)
m
6
=
H:
By denition of multiplication in a factor group, we need to nd
n
so that
g
n
H
=
H
and if
1
m < n
,
g
m
H
6
=
H
