Math 561 H
Fall 2011
Homework 1
Drew Armstrong
1.
Let
G
be a group. Prove that the identity element of
G
is unique (that is, there is only
one element of
G
satisfying the defining property of an identity element). This justifies our
use of the special symbol “
e
”.
2.
Let
G
be a finite group.
(a) Show that there are an odd number of
x
∈
G
such that
x
3
=
e
.
(b) Show that there are an even number of
x
∈
G
such that
x
2
6
=
e
.
(Hint: What is the inverse of
x
n
?)
3.
Let
G
be a finite group.
For all
a, b
∈
G
, show that
ab
and
ba
have the same order as
elements of
G
.
4.
Let
G
be a group and fix an element
g
∈
G
. Define a function
φ
g
:
G
→
G
by
φ
g
(
h
) =
ghg

1
for all
h
∈
G
.
(a) Prove that
φ
g
is a bijection (onetoone and onto).
(b) Prove that
φ
g
(
ab
) =
φ
g
(
a
)
φ
g
(
b
) for all
a, b
∈
G
.
These two properties mean that
φ
g
is an
automorphism
(a “symmetry”) of
G
.
5.
Suppose that 1
,
9
,
16
,
22
,
53
,
74
,
79
,
81 are eight members of a nineelement subgroup of
(
Z
/
91
Z
)
×
. Which element has been left out? Recall: (
Z
/
91
Z
)
×
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 Spring '11
 Armstrong
 Math, Linear Algebra, Algebra, General linear group

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