Math 4107 Study Sheet for the Final Exam
April 26, 2011
1. Know the basics of set theory, mappings, and properties of the integers,
such as divisibility, gcds, the fundamental theorem of arithmetic, prime
numbers, and congruences.
Know how to prove that mappings are
injective, surjective, and bijective.
Know that
f
is invertible iff
f
is
a bijection, as well as how to prove this.
Know that compositions
of surjections are surjections, compositions of injections are injections,
and compositions of bijections are bijections. Know how to show that
if
f
:
X
→
Y
is a bijection, then there is a bijection from
S
X
→
S
Y
,
where
S
A
denotes the set of bijections from
A
→
A
.
2. Know the definition of a group, and how to prove that
G
is a group.
Know some examples of groups, such as
S
n
,
D
n
,
Z
n
, and matrix groups.
Know some examples of nonabelian groups, such as
S
n
,
D
n
and matrix
groups. Know how to construct a group of order
p
3
that is nonabelian,
where
p
is a prime – basically, take 3
×
3 matrices in
Z
p
with 1’s on the
diagonal, 0’s below the diagonal, and arbitrary
Z
p
elements above the
diagonal.
3. Know the definition of a subgroup, and know how to quickly prove that
H < G
is a subgroup of a group
G
. If
G
is finite you only need to check
closure; that is,
h
1
, h
2
∈
H
implies
h
1
h
2
∈
H
. If
G
is not finite, you
need to check that
H
contains the identity and that
h
1
∈
H
,
h
2
∈
H
implies
h
1
h

1
2
∈
H
. Know the “one step subgroup test”: If
a, b
∈
H
implies
ab

1
∈
H
, and
H
is nonempty, and
H
is a subset of a group
G
, then
H
is a subgroup of
G
.
4. Know that if
H, K < G
, then

HK

=

H

K

/

H
∩
K

, as well as the
fact
HK < G
if and only if
HK
=
KH
.
1
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5. Know Lagrange’s theorem, and know how to apply it to show that

H

G

, as well as to prove Euler’s theorem that if
a
is an integer
coprime to
n
(meaning it has no common factors with
n
), then
a
ϕ
(
n
)
≡
1
(mod
n
). Know how to prove that a relation is an equivalence relation.
Know about cosets, and the index of a subgroup
H
in
G
, denoted by
[
G
:
H
].
6. Know normal subgroups. Know how to construct quotient groups.
7. Know homomorphisms and isomorphisms of groups, and how to prove
that maps are homomorphisms and isomorphisms.
8. Know how to show and use the fact that normal subgroups can be
realized as kernels of homomorphisms, and the fact that kernels are
normal subgroups.
9. Know Cayley’s theorem, which says that every group
G
can be embed
ded into a subgroup of a symmetric group. Recall that the symmetric
group
S
on a set
X
, which I denote by
S
X
, is the set of all bijections
ϕ
:
X
→
X
. The proof of Cayley’s theorem amounts to taking
X
as
the elements of the group, and then bijections
ϕ
:
X
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 Fall '08
 Staff
 Algebra, Set Theory, Group Theory, Congruence, Integers, Normal subgroup, Abelian group, UFD, normal subgroups

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