REVIEW SHEET FOR MATH 411 FINAL EXAM, FALL 2011
c 2011 BY SIMAN WONG. ALL RIGHTS RESERVED.
General comments about your exams:
•
TWOHOUR final exam on
Wednesday Dec 14, 1:303:30pm in
LGRC A201
•
no book/notes/calculator (not that calculator will help!)
•
Show your work
, and use complete English sentences!
•
be prepared to recite definitions as well as
precise
statements of theorems
•
the final exam is
cumulative
– you need to know materials covered on both Midterms
and
all of your problem sets!
Permutations
•
for any nonempty set
S
, a
permutation
on
S
is a bijective set function from
S
to itself
•
composition of bijections is a bijection, and the inverse of a bijection is also a bijection, so
Perm(
S
), the set of permutations on the set, forms a group under composition
•
if
S
is a finite set of size
n
then #Perm(
S
) =
n
!
•
in the special case where
S
=
{
1
, . . . , n
}
, we write
S
n
for Perm(
S
); it is called the
permu
tation group on
n
letters
•
if
n
≤
3 then
S
n
is isomorphic to
D
n
, but that is false for
n >
3
Properties of permutations
•
Permutations in
S
n
have two different decompositions:
(1) as a product of
disjoint
cycles
–
this decomposition is
unique
up to reordering
–
this decomposition allows us to inverse of a permutation, compose two permuta
tions, and compute the order of an element in
S
n
; e.g. if
σ
1
, . . . , σ
m
are
pairwise
disjoint
then
(1)
ord(
σ
1
· · ·
σ
m
) =
LCM
(ord(
σ
1
)
, . . . ,
ord(
σ
m
))
.
the proof of this useful fact make uses of, among other things, the crucial prop
erty that
disjoint permutations in
S
n
commute
.
Here is a concrete example: consider the permutation (13)(514)(23)(51):
1
→
5
→
5
→
1
→
3
3
→
3
→
2
→
2
→
2
2
→
2
→
3
→
3
→
1
(132)
4
→
4
→
4
→
5
→
5
5
→
1
→
1
→
4
→
4
o
(45)
Thus (13)(514)(23)(51) = (132)(45), so by (1), we get ord
(
(13)(514)(23)(51)
)
= 6.
(2) as a product of 2cycles
–
thanks to the disjoint cycle decomposition, it suffices to do this for a cycle:
(
a
1
· · ·
a
r
) = (
a
r
a
r

1
)
· · ·
(
a
r
a
1
)
–
this decomposition is
not unique
, but it is a crucial fact that the
parity
of the
number of 2cycles involved is welldefined (in other words: the number of 2
cycles involved modulo 2). This allows us to declare that an element in
S
n
is
odd
or
even
depends on whether the number of 2cycles involved is odd or even.
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c 2011 BY SIMAN WONG. ALL RIGHTS RESERVED.
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 Winter '11
 PhanThuongCang
 Symmetric group, finite group, SIMAN WONG

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