MATH 212 algebra

Group Axioms

Closure

Associativity

Identity

Inverse

Ring Axioms

1. (R, + ) is an abelian group

2. (R, * ) is associative

3. * distributes over +

Ring with unity or 1

A ring with the usual axioms as well as having a multiplicative identity

a*1 = 1*a = a

1. (R, + ) is an abelian group

2. (R, * ) is associative

3. * distributes over +

4. (R, * ) has an identity

Commutative Ring

A ring with the usual axioms as well as being commutative with *

a*b = b*a

1. (R, + ) is an abelian group

2. (R, * ) is associative

3. * distributes over +

4. (R, * ) has an identity

5. (R, * ) is commutative

Unit for a ring

an element of a ring that has a multiplicative inverse

a*a^-1 = 1

Integral Domain

non-trivial commutative ring with unity that has no zero divisors

always a sub-ring of a field

finite integral domains are always fields

1. (R, + ) is an abelian group

2. (R, * ) is associative

3. * distributes over +

4. (R, * ) has an identity

5. (R, * ) is commutative

6. a*b = 0 ==> a or b = 0

Field

A commutative ring with unity in which 1≠0 and all non zero elements are units

1. (R, + ) is an abelian group

2. (R, * ) is associative

3. * distributes over +

4. (R, * ) has an identity

5. (R, * ) is commutative

6. a*b = 0 ==> a or b = 0

7. (R, * ) has multiplicative inverses for all nonzero elements

Orbit

All the elements that a given element from a set S can be mapped to by the actions from a group G.

It will be a subset of S.

G(x) = {g(x): g∈G}

Stabilizer

All the actions from a group G that map a given element from a set S to itself

It will be a subgroup of G.

G_x = {g∈G: g(x) = x}

Orbit Stabilizer Theory

|G| = |G_x|*|G(x)|

properties of a stabilizer G_x

• Composing symmetries in G_x stays in G_x

• Identity is in G_x

• Undoing any symmetry in G_x stays in G_x

• G_x is a subgroup of G

• |G_x| divides |G|

Subgroup Conditions

if H is a subgroup of G

1. H is non-empty

2. For any elements a, b ∈ H, ab^-1 ∈ H

Lagrange’s Theorem

for a subgroup H of G, |H| divides |G|

ie, the order of every subgroup of G divides the order of G

Let H be a subgroup of G, and let G have order k. What could the order of H be?

|H| must be a positive integer that divides |G|

Let H be a subgroup of G, and |G| is prime. What can we infer about H and G?

G is a cyclic group, and H = {e} or G

In any group, (ab)^-1 = ____

b^-1a^-1