Alg Flashcards
standard sets
1
Z,Q,R,C,M_{mxn}(S), M_n(S)”+” superscript, “+” with overbar, *
binary operation on S
2
closed under an operation
3
associative
4
semigroup
5
identity
6
monoid
7
inverse
8
group
9
defn, equivalently 4 conditions
properties of groups
10
6 of em
commutative
11
Abelian
12
order of S
13
examples of groups
14
7 groups, 2 more
general linear group
15
special linear group
16
nth roots of unity
17
symmetric group on n letters
18
dihedral group
19
group table
20
additive group of integers modulo n
21
multiplicative group of integers modulo p
22
cyclic group
23
generator of (Z_n,+)
24
when a^m is a generator of G
25
subgroup
26
center
27
cyclic subgroup generated by a
28
order of an element
29
finitely generated group
30
relations
31
defn, Klein four-group presentation
Lagrange’s theorem
32
Abelian subgroup of certain order
33
subgroups of cyclic groups
34
Cauchy’s theorem
35
Sylow’s first theorem
36
isomorphic
37
direct product
38
when Z_m+Z_n+… is cyclic
39
elementary divisors of G
40
invariant factors of G
41
coset
42
group homomorphism
43
monomorphism
44
epimorphism
45
isomorphism
46
endomorphism
47
automorphism
48
properties of homomorphisms
49
6 of em
kernel
50
ring
51
3 main conditions, total of 8 sub-conditions
ring with unity
52
commutative ring
53
subring
54
characteristic of the ring
55
ring of integers
56
ring of integers modulo n
57
ring of Gaussian integers
58
ring of real-valued functions on R
59
ring homomorphism
60
properties of ring homomorphism
61
3 of em
integral domain
62
zero divisor
63
zero divisors in Z_n
64
integral domains Z_n
65
unit
66
division ring
67
field
68
defn, 6 conditions
