Abstract: Groups

Group properties

• associative under bin op

• there’s an identity element in G (unique)

• every element has an inverse in G (unique inverse)

• $$\left(a\cdot b\right)^{-1}=b^{-1}\cdot a^{-1}$$

Notation for commutative binary operations

$a \cdot b = a + b$ and identity is 0

Subset Theorem

For subset H in G, H is a subgroup if and only if

1. H is closed under inverses, if $h \in H \rightarrow{} h^{-1} \in H$

2. H is closed under multiplication, if $h_1, h_2 \in H \rightarrow{} h _1^{-1} h_2^{-1} \in H$

Center of a Group $Z(G)$

Set of elements that commute with all other elements in G

Nonempty Subset Theorem

Let $H \subseteq G$ be nonempty, then $H \leqslant G$ iff $\forall a,b \in H, ab^{-1} \in H$

Note: a and b can be equal

Set of generators

$\forall g \in G$, then elements can be written as a finite product of elements of $S \subseteq G$ and their inverses; $<S>$

Ex: Set of symmetries of the equilateral triangle $D_6$ can be rewritten as the product of $\rho_1, \mu_1$ making those elements a set of generators (dihedral & sym notes)

Relations

An equation made from the elements in $S$

Ex: $\rho³ = 1$ where 1 is the identity of $S$

Presentation of a group

A different way to represent $G$, consists of a set of generators and relations

$$G = <S|R_1, R_2, …,R_m>$$

Dihedral group of order 2n

A group of symmetries of a regular polygon with n vertices

$$D_{2n} = <r,s | r^n = s²=1, rs = sr^-1>$$

Image

For a function that sends elements of A to elements of B, the image is the range of that f sends to elements of A

$$f(A) = {f(a) \in B | a \in A}$$

Injective (one-to-one)

when $f(a) = f(a’) \ \text{implies} \ a = a’$ (proof technique), meaning that the input can only map to one element in the codomain, the range is not the entire domain

Note: The kernel for injective only contains the identity element.

Surjective (onto)

When the range of f(A) = B, in other words, all elements in B is covered and has at least one element (can be more than one) in domain that maps to it.

Proof technique is to show that all elements come from somewhere.

Invertible map remark

A map is invertible if it is bijective

Cycles

Let $\sigma \in S_n. \ \sigma$ is a cycle of length $k$ or a k-cycle if there exists k elements in the cycle. Applying \sigm moves an element forward in the cycle. And fixes all other elements.

Disjoint cycles

Two cycles are disjoint if if they have no numbers in common, they also commute

Order of a cycle

$k$-cycle has order $k$

Cycle decomposition

Every permutation in $S_n$ can be written as a product of disjoint cycles called a cycle decomposition.

The order of any permutation is the least common multiple of the lengths of the cycles in its cycle decomposition

The identity map of $G$

Map that maps an element to itself

$$\vec{1}_G: G \rightarrow G \ s.t \ \vec{1}_G(x) = x$$

Trivial Homomorphism

A map that maps G to G such that $\phi(x) = \bold{1}$, sends all elements to the identity

Properties of homomorphism

• $\phi(\bold{1}) = \bold{1}$ identities map to identities

• $$\phi(x^{-1}) = (\phi(x))^{-1}$$

• $$\phi(x_1, . . ., x_n) = \phi(x_1). . . \phi(x_n)$$

• If there’s a another mapping, the transitivie property applies if possible

Isomorphism properties

• $$|G| = |H|$$

• $G$ is abelian iff $H$ is abelian

• $$\forall x \in G, |x| = |\phi(x)|$$

Inner automorphism

$a \in G$ let $\alpha_a: G \rightarrow G$ where $\alpha_a(x) = axa^{-1}$ , we can define $\alpha: G \rightarrow \text{Aut}(G)$ with $\alpha(a) = \alpha_a$

Cosets

Let $H \le G \ \text{and let} \ x \in G \text{then} Hx = {hx | h \in H}$ .

Remark: If $G$ is abelian, then we write $h + x$ (this is right coset, same applied to the left coset)

Two right (or left) cosets are either identical or disjoint

$xRx$ implies $xy^{-1} \in H$

Number of right cosets of a subgroup equal the number of left cosets

Index of $H$ of $G$

$[G : H] =$ number of distinct cosets (right/left)

Lagrange Theorem

If H is a subsgroup of G, then $|G| = |H|[G:H]$

Any two cosets have the same cardinality $|H| = |Hx| = |Hy|$

If G is finite:

• order of every subgroup of G divides order of G

• index of every subgroup divides order of G

• order of every element divides G

  • in group $\Z_n$, if n is prime, the every non-identity element is that prime number

• $$a^{|G|} = \bold{1}$$