Modern Algebra

• when the order does not matter
• always get the same result no matter which order we do the binary operation

• a set G with a binary operation *, (G, *) such that:
• 1. the operation * is closed and well-defined on G
• 2. The operation * is associative on G
• 3. There is an element e in G such that g*e = g = e*g for all elements g in G.  e is called the identity. G is non-empty.
• 4. for each element g in G there is an element h in G such that g*h = e = h*g.  the inverse of g, g^-1.

there exists an element e in G such that g * e = g = g * e for all g in G.

• g^-1
• g * h = e = h * g where h is the inverse of g.

The integers with addition, (Z, +), is a group

The nonzero real numbers with multiplication (R{0}, .) is a group.

• the transformations of an equilateral triangle in the plane with composition is a group.
• (D₃,) the symmetries of the equilateral triangle

• Let G be a group.
• There is a unique identity element in G.
• There exists only one element in G, e, such that g * e = g = g * e for all g in G.

• Let G be a group and let a, x, y exist in G.
• Then a * x = a * y iff x = y.

• Let G be a group
• Then each element g in G has a unique inverse in G.
• There is only one element h such that g * h = e = h * g.

• Let G be a group with elements g and h.
• If g * h = e, then h * g = e.

• Let G be a group and g be in G.
• Then (g^-1)^-1 = g.

• Fore every natural number n, the set C_n with n-cyclic addition, (C_n, _n) is a group.
• Called the cyclic group of order n.

a transformation that takes the regular polygon to itself as a rigid object.

The symmetries of the square in the plane with composition form a group.

The symmetries of a regular n-gon form a group, denoted D_n

• a = b mod n
• two integers iff their difference is divisible by n
• a and b are congruent modulo n if there exists an integer k st a = b+kn or a-b = kn

• let Z_n = {[a]_n| a Z, [a]_n = [b]_n iff a = b mod n}
• on Z is defined by [a]_n [b]_n = [a+b]_n

• a subgroup of a group (G, *) is a non-empty subset, H, of G along with the restricted binary operation such that (H, *|H) is a group
• to check that a subset is a subgroup, must check all conditions of the group

• Let G be a group with identity element e.
• Then for every subgroup H of G, e is in H.

• Let G be a group with identity element e.
• Then {e} is a subgroup of G.

• Let G be a group.
• Then G is a subgroup of G.

• {e} and G are trivial
• if H is a subgroup of G and not either of these it is non-trivial.

• g*g*g*g
• repeated operations of the binary operation to one element g in G.
• g⁰=e
• g¹=g

• subset of elements of G formed by repeated operations using only g and g^-1
• <g> = {g^+-1*g^+-1*g^+-1...}

• Let G be a group and g be an element of G.
• Then <g> is a subgroup of G.

• Let G be a group and S be a subset of G.
• Then <S> is a subgroup of G.

• a group G is called cyclic if there is an element g in G st <g> = G.
• A group is cyclic if it is generated by one element

The integers under addition is a cyclic group

Any subgroup of a cyclic group is cyclic

• The groups D_n for n>2 are not cyclic.
• Need flips and rotations.

a group G is finite if the underlying set is finite

a group G is infinite if the underlying set is infinite

G = <S> for some finite subset S

Every finite group G is finitely generated.

• the number of elements in G, |G|.
• the order of an element g, o(g) is the order of the subgroup that it generates.
• o(g) = |<g>|

• a group (G, *) is abelian iff for every pair of elements g, h in G, g*h=h*g.
• iff its binary operation is commutative

• the center of a group is the collection of elements in g that commute with all elements of G
• Z(G) = {g G | g*h = h*g V h G}

• Let G be a group
• Then Z(G) is a subgroup of G.

• Let H be a subgroup of group G and g be and element of G.
• The left coset of H by g is the set of all elements of the form gh for all h that exist in H.
• written as gH = {gh | h  H}

Let H be a subgroup of G and let g and g' be elements of G.  Then the cosets gH and g'H are either identical or disjoint.

Let G be a finite group with subgroup H.  Then |H| divides |G|.

Let G be a finite group with a subgroup H. Then the number of left cosets of H is equal to the number of right cosets of H.

• Let H be a subgroup of a group G.  Then the index of H in G is the number of distinct left or right cosets of H. 
• Written as [G : H]

Let G be a finite group with a subgroup H.  Then [G : H] = |G|/|H|.

Let G be a finite group with an element g.  Then o(g) divides |G|.

If p is a prime and G is a group with |G| = p, then G has no non-trivial subgroups.

If A and B are sets, then we define A x B = {(a,b)| a  A and b  B}, the set of ordered pairs of elements from A and B.

• Let (G, *_G) and (H, *_H) be groups and define *: (G x H) x (G x H)  G x H by (g₁, h₁) * (g₂, h₂) = (g₁ *_G g₂, h₁ *_H h₂). 
• Then (G x H, *) is a group, called the (direct) product of G and H.

• Let G and H be groups.
• Then G x H is abelian iff both G and H are abelian.

Let G₁ be a subgroup of a group G and H₁ be a subgroup of a group H. Then G₁ x H₁ is a subgroup of G x H.

(123)

• Let X be a set, let Sym(X) be the set of bijections from X to X, and let  represent composition.  Then (Sym(X), ) is a group.
• This is the symmetric group on X.

• Let (G, *_G) and (H, *_H) be groups and let  : G  H be a function on their underlying sets. 
•  is called a homomorphism of the groups if for every pair of elements g_1, g₂  G,  (g₁ *_G g₂) = (g₁) *_H  (g₂).

• iAB : A  B is defined as follows:
• for each element a  A, iAB(a) = a.

• Let A and B be sets and f : A  B be a function.  For any subsets S  A and T  B we define
• 1. Im_f(S) = {b  B | there exists an a  S st f(a) = b}.  We call this the image of S under f.
• 2.Preim_f(T) = {a  A | f(a)  T}.  We call this the preimage of T under f.

Let  : G  H be a homomorphism from a group G to a group H. Then the set {g  G |(g) = e_H} is called the Kernel and is denoted Ker().

an injective homomorphism

a bijective homomorphism

A group G is isomorphic to a group H if there exists an isomorphism,  : G  H.