deck 1 Flashcards
(102 cards)
1
Q
what are the three conditions of a binary operation?
A
-
2
Q
what’s is the definition of a binary operation ?
A
a rule that assigns to each ordered pair of elements of S a uniquely determined element of S
3
Q
define a Group
A
-
4
Q
what are the three conditions to identify a group
A
associative, identity, inverse
5
Q
what binary operation Z_n under?
A
addition
6
Q
what binary operation is Z_n{0} under
A
x
7
Q
define the symmetric group and what binary operation is it under
A
the set of all permutations. composite
8
Q
define abelian
A
if the binary operation of G is commutative, than G is abelian
9
Q
is Z_n abelian?
A
yes
10
Q
is S_n abelian?
A
no when n>3
11
Q
is GL(n,R) abelian ?
A
no when n>2
12
Q
define GL(n,R), what is it a group under
A
the set of all invertible matrices. matrix multiplication
13
Q
what is the order of a group ?
A
the number of elements in the set G
14
Q
|Z|= ?
A
infinity
15
Q
|Z_n|= ?
A
n
16
Q
|S_n|= ?
A
n!
17
Q
|GL(n,R)|= ?
A
infinity
18
Q
the set M_n(R) is a group under ?
A
addition
19
Q
define a subgroup
A
H<G, and H is closed under a binary operation
20
Q
what are the 4 conditions for a subgroup
A
closure, associative, inverse, identity
21
Q
define cyclic subgroup <a></a>
A
-
22
Q
is <a> abelian ?</a>
A
yes
23
Q
define the order of a purmutation
A
the smallest natural number m, such that σ^m = e
24
Q
the order of a cycle is equal to ?
A
its length
25
the order of a disjoint cycle is equal to
the LCM of its lengths
26
define the order of an element
the m such that a^m =e
27
lemma 3.4. if a has finite order n.
| than elements a,a^2,a^3... are ____?
distinct
28
define the special linear group
-
29
define the scalar group
-
30
define the diagonal group
-
31
define the triangular group
-
32
define the upper triangular
-
33
|D(n,Z_p)| = ?
-
34
|UTu(n,Z_p)| = ?
-
35
define the centraliser C(a)
the set of all elements that are commutative
37
any group C(e) = ?
G
38
any abelian group C(a)= ?
G
39
define the centre of G. Z(G).
the set of all elements that commute with all other elements in G
40
C(12...n) = ? and recite the proof for it
41
any abelian group Z(G)=?
G
42
Z(a)< ?
C(a)
43
Z(S_n) = ?
{e}
44
Z(GL(n,R)) =
scal(n,R)
45
define a cyclic group
-
46
define a generator of G
G=cyclic if all elements are powers of a single a. ais a generator
47
if G is cyclic and |G|=n, H
divides n
50
define left coset
H
51
H
xH=yH
52
define the index [G:H]
the number of distinct left cosets of h in G
53
|gH|= ?
|H|
54
state lagranges theorem.
|G|=n. H
55
if H
divides |G|
56
if g is in G, the order of g does what?
divides |G|
57
if |G| =p what is G
cyclic
58
R is a group under? and what is its identity?
addition e=0
59
R* is a group under? and what is its identity
x, e=1
60
C is a group under? and what is its identity?
+, e=0
61
C* is a group under? and what is its identity?
x, e=1
62
state the set S_2
-
63
state the set S_3
{(12),(23),(13),(123),(132)}
64
define a homomorphism
-
65
what should you look out for when doing homomorphisms ?
the binary operations can be different, it depends on which groups your function is over.
66
what are the three properties of a homomorphism
1. maps e of G to e of H
2. ψ(a^-1) = ( ψ(a) )^-1
3. im(ψ) = subgroup of H. Imψ < H
67
define an isomorphism
a homomorphism that is 1-1
68
what are group theoretic properties, and what can you use them for.?
they are common to all isomorphic groups. and can be used to prove that two groups are not isomorphic
69
give some examples of a group theoretic property
being- finite, infinite, abelian, cyclic. having an element of finite order, element of order n, trivial centre, non trivial centre
70
define conjugacy
b is a conjugate of a if b=xax^-1
71
what is the set notation for a conjugate element
-
72
e^G =
{e}
73
a^G ={a} iff ?
a belongs to Z(G)
74
|e^G| = ?
1
75
conjugate permutations are similar in what way
they have the same cycle structure
76
define a partition
a natural number n is a sequence of natural numbers.
| ie 4=2+1+1
77
define the class formula
the number of conjugates of an element is equal to the index of the centralizer of a in G
78
define a normal subgroup, and what's the notation
H
79
define the kernel of a homomorphism
is the subset of G defined by the set of all elements in G that under the homomorphism create e
80
the ker(ψ) is a normal subgroup of ?
G
81
Z(G) is a normal subgroup of
G
82
any two subgroups of index 2 in G are ?
normal
83
define factor group
the set of all left cosets of N in G
84
xNyN = ?
xyN
85
define the natural isomorphism
a homomorphism from G---> G\N.
| ie from G to the factor group
86
define the first isomorphism
?
87
if a multiplication table is symmetric, what does this impliyt
the binary operation is commutative
88
how to you work out the order of a permutation
it is the LCM of the disjoint cycles
89
how do you work out a centraliser of a matrix g ?
matrix multiply by a default matrix, with values a,b,c,d.
90
if τ is conjugate to σ, how do you find θ such that τ=θσθ^-1
write τ below σ, so that there same disjoint cycles are beneath one another. then simply write θ as the each element corresponds to the one below.
91
how do you find the number of conjugacy classes in S_n
write n as all its partitions, the number of possible partitions is the number of conjugacy classes
92
Prove C((12...n)) =
-
93
method to find y such that τ = yσy^-1
put σ on top and τ below so there cycle structure match up, then y equals the corresponding elements above and below each other.
94
to work out the left coset, do what?
each element individually
95
define a conjugate and a conjugacy class
```
b is conjugate to a if b=xax^-1 for some x in G.
a conjugacy class is the set of all conjugates of a.
```
96
in the symmetric group, what condition is necessary for two cycles to be conjugate
they have to have the same cycle structure
97
how do you work out σ^(S_n) ? conjugacy class
all the elements if S_n that have the same cycle structure as σ
98
list the S_4 group
-
99
lagranges thm ?
|G| = [G:H]|H|
100
what helps when working out C(σ) in S_n?
- the inverse of σ could be commutative
- all the cycles with the elements not in σ
- combination of the above two
- and of course, e
101
define the factor group, what binary operation is it under
(x)N(y)N = (xy)N
102
what is it the property of left cosets, when x,y are in G and H
either they are the same or completely different.
| xH=yH or xH n yH = empty set
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