NoteNextra-origin/content/Math4302/Exam_reviews/Math4302_E1.md

Math 4302 Exam 1 Review

Note

This is a review for definitions we covered in the classes. It may serve as a cheat sheet for the exam if you are allowed to use it.

Groups

Basic definitions

Definition for group

A group is a set G with a binary operation * that satisfies the following axioms:

1. Closure: \forall a,b\in G, a* b\in G (automatically guaranteed by definition of binary operation).
2. Associativity: \forall a,b,c\in G, (a* b)* c=a* (b* c).
3. Identity: \exists e\in G, \forall a\in G, e* a=a* e=a.
4. Inverses: \forall a\in G, \exists a^{-1}\in G, a* a^{-1}=a^{-1}* a=e.

• Identity element: If X has an identity element, then it is unique.
• Composition of function is associative.

Order of a element

The order of an element a in a group G is the size of the smallest subgroup generated by a, we denote such subgroup as \langle a\rangle.

Equivalently, the order of a is the smallest positive integer n such that a^n=e.

Order of a group

The order of a group G is the size of G.

Definition of subgroup

A subgroup H of a group G is a subset of G that is closed under the group operation. Denoted as H\leq G.

Left and right cosets

If H is a subgroup of G, then aH is a coset of H for all a\in G. We call aH a left coset of H for a.

aH=\{x|a\sim x\}=\{x\in G|a^{-1}x\in H\}=\{x|x=ah\text{ for some }h\in H\}

Similarly, Ha is a right coset of H for a.

Ha=\{x|x\sim'a\}=\{x\in G|xa^{-1}\in H\}=\{x|x=ha\text{ for some }h\in H\}

• Usually, the left coset and right cosets will give different partitions of G.
• Use to prove lagrange theorem (partition of G into cosets)

Definition of normal subgroup

A subgroup H of a group G is normal if aH=Ha for all a\in G.

Isomorphism and homomorphism

Definition of isomorphism

Two groups G and G' are isomorphic if there exists a function f:G\to G' such that

• Homomorphism property is satisfied: f(a*b)=f(a)f(b),\forall a,b\in G
• f is injective: f(a)=f(b)\implies a=b
• f is surjective: \forall a\in G',\exists b\in G such that f(b)=a

Definition of homomorphism

A homomorphism is a function that satisfies the homomorphism property.

If \phi:G\to G' is a homomorphism, then

• \phi(e)=e', where e is the identity of G and e' is the identity of G'.
• \phi(a^{-1})=(\phi(a))^{-1} for all a\in G.
• If H\leq G is a subgroup, then \phi(H)\leq G' is a subgroup.
• If K\leq G' is a subgroup, then \phi^{-1}(K)\leq G is a subgroup.
• \phi is surjective if and only if \operatorname{ker}(\phi)=\{e\} (the trivial subgroup of G).

Basic groups

Trivial group

The group (\{e\},*) is called the trivial group.

Abelian group

A group G is abelian if a*b=b*a for all a,b\in G.

• The smallest non-abelian group is S_3 (order 6).
• Every abelian group is isomorphic to some direct product of cyclic groups of the form:

  
  \mathbb{Z}_{p_1^{n_1}}\times \mathbb{Z}_{p_2^{n_2}}\times \cdots \times \mathbb{Z}_{p_k^{n_k}}\times\underbrace{\mathbb{Z}\times \ldots \times \mathbb{Z}}_{m\text{ times}}
  

Cyclic group

A group G is cyclic if G is a subgroup generated by a\in G. (may be infinite)

• The smallest non-cyclic group is Klein 4-group (order 4).
• Every group with prime order is cyclic.
• Every cyclic group is abelian.
• If G has order n, then G is isomorphic to (\mathbb{Z}_n,+).
• If G is infinite, then G is isomorphic to (\mathbb{Z},+).
• If G=\langle a\rangle and H=\langle a^k\rangle, then |H|=\frac{|G|}{d} where d=\operatorname{gcd}(|G|,|H|).
• Every subgroup of cyclic group is also cyclic.

Dihedral group

The dihedral group D_n is the group of all symmetries of a regular polygon with n sides.

• |D_n|=2n.
• It is finitely generated by \{\rho,\phi\}, where \rho is a rotation of a regular polygon by \frac{2\pi}{n}, and \phi is a reflection of a regular polygon with respect to $x$-axis.

Symmetric group

The symmetric group S_n is the group of all permutations of n objects.

• S_n has order n!.
• Every group G is isomorphic to S_A for some A.
• Odd and even permutations
  • Every permutation can be written as a product of transpositions.
  • A_n is the alternating group with order \frac{n!}{2} consisting of all even permutations.
  • A non trivial homomorphism from S_n to (\Z_2,+) is given by \sigma\mapsto \begin{cases} 0 & \text{if } \sigma\text{ is even} \\ 1 & \text{if } \sigma\text{ is odd} \end{cases}