Definitions Flashcards
Lagrange’s Theorem
Let G be a finite group and H be a subgroup of G. The # of G is a divisor of #H
The size of a subgroup of a finite group is a divisor of the size of that group
Group
A set G with a binary operation G x G -> G, written as (g,h) -> gh, satisfying three axioms
1. Existence of Identity
2. Associative Law
3. Existence of Inverse
Abelian Group
let G be a group, with the binary operation *, G is called an abelian group(commutative group) if, for all elements of a,b in G if a * b = b *a
Subgroup
Let G be a group. A subset H of G is a subgroup of G, if it has the following properties:
1. Existence of Identity
2. Associative Law
3. Existence of Inverse
left congruence
Equivalance classes of the left cosets H g
Right Congruence
Equivalance classes of the right cosets H g
Normal subgroup
Left and right congruence modulo H are the same equivalence relation
Homomorphism between group
Let G and H be groups with binary operations, *g and *h, respecting. A homomorphism from G to H is a function f: G -> H such that for all elements g1, g2, in G, the following property hold:
f(g1 *G g2) = f(g1) *H f(g2)
Isomorphomism between groups
Invertible homomorphism, two groups are essentially the same in terms of their structure, even if their elements and operations look different.
Quotients of a group by a normal subgroup
G/H is a set of cosets of H of G. The binary operation makes R a group that:
1. Multiplication is associative
2. There is an identity element
3. Multiplication distributes over addition
Ring
A set R with the following structures:
1. Binary operations
2. Identity element
e. Inverse operation
Commutative Ring
Ring in which multiplication is commutative
Unit in a ring
The set of R that groups R^x under multiplication
Field
a type of commutative ring in which every nonzero element has a multiplicative inverse
