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

group
 a set G with a binary operation *, (G, *) such that:
 1. the operation * is closed and welldefined 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 nonempty.
 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}.

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

inverse
 g^{1}
 g * h = e = h * g where h is the inverse of g.

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

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

Theorem
 the transformations of an equilateral triangle in the plane with composition is a group.
 (D_{3},) the symmetries of the equilateral triangle

Theorem 3.6  unique identity
 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.

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

Corolary 3.9  unique inverse
 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.

Theorem 3.10  identity is commutative
 Let G be a group with elements g and h.
 If g * h = e, then h * g = e.

Theorem 3.11  h and g are inverses
 Let G be a group and g be in G.
 Then (g^{1})^{1} = g.

Theorem 3.13  cyclic group of order n
 Fore every natural number n, the set C_{n} with ncyclic addition, (C_{n}, _{n}) is a group.
 Called the cyclic group of order n.

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

Theorem 3.19  symmetries of the square
The symmetries of the square in the plane with composition form a group.

dihedral group
The symmetries of a regular ngon form a group, denoted D_{n}

congruent modulo
 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 ab = kn


subgroup
 a subgroup of a group (G, *) is a nonempty 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

Theorem 3.23  identity in subgroup
 Let G be a group with identity element e.
 Then for every subgroup H of G, e is in H.

Theorem 3.24  identity is a subgroup of G
 Let G be a group with identity element e.
 Then {e} is a subgroup of G.

Theorem 3.25  group is subgroup of itself
 Let G be a group.
 Then G is a subgroup of G.

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

g^{4}
 g*g*g*g
 repeated operations of the binary operation to one element g in G.
 g^{0}=e
 g^{1}=g

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

Theorem 3.27  subgroup of G generated by g
 Let G be a group and g be an element of G.
 Then <g> is a subgroup of G.

Theorem 3.29
 Let G be a group and S be a subset of G.
 Then <S> is a subgroup of G.

Cyclic
 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

Theorem 3.31  integers under +
The integers under addition is a cyclic group

Theorem 3.33  cyclic has cyclic subgroups
Any subgroup of a cyclic group is cyclic

Theorem 3.34  D_{n} not cyclic
 The groups D_{n} for n>2 are not cyclic.
 Need flips and rotations.

Finite
a group G is finite if the underlying set is finite

Infinite
a group G is infinite if the underlying set is infinite

Finitely Generated
G = <S> for some finite subset S

Theorem 3.36  finite group is finitely generated
Every finite group G is finitely generated.

order
 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>

Abelian
 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

center
 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}

Theorem 3.45
 Let G be a group
 Then Z(G) is a subgroup of G.

Left coset of H by 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}

Lemma 3.48
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.

Lagrange's Theorem 3.49
Let G be a finite group with subgroup H. Then H divides G.

Scholium 3.50
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.

Index
 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]

Scholium 3.51
Let G be a finite group with a subgroup H. Then [G : H] = G/H.

Corollary 3.52
Let G be a finite group with an element g. Then o(g) divides G.

Corollary 3.53
If p is a prime and G is a group with G = p, then G has no nontrivial subgroups.

Cartesian Product
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.

Theorem 3.54
Direct Product
 Let (G, *_{G}) and (H, *_{H}) be groups and define *: (G x H) x (G x H) G x H by (g_{1}, h_{1}) * (g_{2}, h_{2}) = (g_{1} *_{G} g_{2}, h_{1} *_{H} h_{2}).
 Then (G x H, *) is a group, called the (direct) product of G and H.

Theorem 3.56
 Let G and H be groups.
 Then G x H is abelian iff both G and H are abelian.

Theorem 3.57
Let G_{1} be a subgroup of a group G and H_{1} be a subgroup of a group H. Then G_{1} x H_{1} is a subgroup of G x H.



Theorem 3.66
Symmetric group
 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.




kernel
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( ).

Monomorphism
an injective homomorphism

isomorphism
a bijective homomorphism

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

