1
Definition of <group>
Set G G ≠{0}
Operation G × G → G ; ( a, b ) → ab ab is called <product>.
Conditions of operation
(1) <associative law> arbitrary a, b, c ∈ G (ab)c = a(bc)
(2)<identity element> e ∈ G arbitrary a ∈ G ae = ea = a also expressed by 1G
(3)<inverse element> e ∈ G arbitrary a ∈ G ab = ba = e also expressed by a-1
Another additional condition of operation
(4)<commutative law> arbitrary a, b ∈ G ab = ba G is called <Abelian group>.
When productive operation is done by <addition>, Abelian group is called <additive group> or <module>.
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Definition of <subgroup>
Group G
Subset H ∈ G
Conditions of H
(1)a, b ∈ H ⇒ ab ∈ H
(2)a ∈ H ⇒ a-1 ∈ H
Definition of <normal subgroup>
Arbitrary h ∈ H g ∈ H
g-1 hg ∈ H
H normal subgroup
3Definition of <finite group>
Group G
G has finite elements.
G finite group
Definition of <order>
G’s finite elements
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Definition of <homomorphism of group>
Map f : G → H
Arbitrary a, b ∈ G
f (ab) = f (a) f (b)
5
Definition of, isomorphism of group>
f is bijective, i.e. f is injective ( map f : A → B a, a’ ∈ A f ( a ) = f ( a’ ) ⇒ a = a’ ) and f is surjective (map f : A → B Image ( f ) = B ).
Expression is G ≅ H
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Definition of <ring>
Additive group A
Product operation of A AA → A ; ( x, y ) → xy
(1) <associative law> (xy)z = x(yz)
(2) <distributive law> (x + y)z = xz + yz z(x + y) = zx + zy
(3) <identity element> e ∈ A x ∈ A xe = ex = x
Definition of <commutative ring>
Another additional condition of operation
(4)<commutative law> xy = yx
Tokyo September 11, 2007
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