The flashcards below were created by user
mi06bian
on FreezingBlue Flashcards.

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.

