Ch 2 + 3 - Groups and Finite Groups; Subgroups

Let G be a set. A binary operation G is

a function that assigns each ordered pair of elements of G an element of G

Z_n

the binary operations are addition modulo n and multiplication mdule n on the set { 0, 1, 2, …, n - 1}

Let G be a set together with a binary operation that assigns to each ordered pair (a, b) of elements of G an element in G denoted by ab. We say G is a group under this operation if the following three properties are satisifed

1. Associativity. The operation is associative; that is, (ab)c = a(bc) for all a, b, c in G

2. Identity. There is an element e (called the identity) in G such that ae = ea = a for all a in G

3. Inverses. For each element a in G, there is an element b in G ( called an inverse of a) such that ab = ba =e.

A group is Abelian if

it has the property that ab = ba for every pair of elements a and b

Z

the set of integers; the identity is 0; the inverse of a is -a

Q

the set of rational numbers; the identity is 0; the inverse of a is -a

R

the set of real numbers the identity is 0; the inverse of a is -a

theorem 2.1 - Uniqueness of the identity

In a group G, there is

only one identity element

theorem 2.2 -- Cancellation

In a group, the right and left cancellation laws holds; that is,

ba = ca implies b = c, and ab and ac implies b = c

theorem 2.3 - Uniqueness of Inverses

For each element a n a group G, there is

a unique element b in G such that ab = ba = e

Theorem 2.4 - Socks -Shoes Property

For group elements a and b,

(ab)^-1 = b^-1 a^-1

|G| - The order of a group is

the number of elements of a group(finite or infinite)

|g| - The order g in a group G is

the smallest positive integer n such that gⁿ = e. (In addition notation, there would be ng = 0.)

H is a subgroup of G if

the subset H of a group G is itself a group under the operation of G

H < G - H is a proper subgroup of G

if H is not equal to G

Trivial subgroup of G

is {e}

Nontrivial subgroup

is a subgroup that is not {e}

Theorem 3.1 - One-step subgroup test

Let G be a group and H be a nonempty subset of G. H is a subgroup of G if

ab^-1 is in H whenever a and b are in H. (IN additive notation, if a - b is in H whenever a and b are in H, then H is a subgroup.)

Theorem 3.2 - Two-Step Subgroup

Let G be a group and let H be a nonempty subset of G. H is a subgroup of G if

ab is in H whenever a and b are in H (H is closed under the operation), and a^-1 is in H whenever a is in H (H is closed under taking inverses)

How to prove that a subset is not a subgroup?

show the identity is not in the set, exhibit an element of the set whose inverse is not in the set, or exhibit two elements of the set whose product is not in the set

Theorem 3.3 - Finite Subgroup Test

Let H be a nonempty finite subset of a group G. H is a subgroup of G if

H is closed under the operation of G

For any element a, we let <a> denote the set

{aⁿ | n belonging to Z}

Theorem 3.4 - <a> is a subgroup

Let G be a group, and let a be any element of G. Then,

<a> is a subgroup of G

<a>

is a cyclic subgroup

If G =<a>

G is cyclic and a is a generator of G

Every cyclic group is

Abelian

when the operation is addition, aⁿ means

na

The center, Z(G), of a group, is the

subset of elements in G that commute with every element of G = {a is an element of G | ax = xa for all x in G}

Theorem 3.5

The center of a group is a

subgroup of G

Let a be a fixed element of a group G. The centralizer of a in G, C(a), is the

set of all elements in G that commute with a; = {g is an element of G| ga = ag}

Theorem 3.6

For each a in a group G, the centralizer of a a is a

subgroup of G

G is Abelian if and only if

C(a) = G for all a in G

For every element a of a group G, Z(G) is a subset of

C(a)