124 lines
4.6 KiB
Markdown
124 lines
4.6 KiB
Markdown
# Math 4302 Exam 1 Review
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> [!NOTE]
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>
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> 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.
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## Groups
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### Basic definitions
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#### Definition for group
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A group is a set $G$ with a binary operation $*$ that satisfies the following axioms:
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1. Closure: $\forall a,b\in G, a* b\in G$ (automatically guaranteed by definition of binary operation).
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2. Associativity: $\forall a,b,c\in G, (a* b)* c=a* (b* c)$.
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3. Identity: $\exists e\in G, \forall a\in G, e* a=a* e=a$.
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4. Inverses: $\forall a\in G, \exists a^{-1}\in G, a* a^{-1}=a^{-1}* a=e$.
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- Identity element: If $X$ has an identity element, then it is unique.
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- Composition of function is associative.
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#### Order of a element
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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 $\langle a\rangle$.
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Equivalently, the order of $a$ is the smallest positive integer $n$ such that $a^n=e$.
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#### Order of a group
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The order of a group $G$ is the size of $G$.
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#### Definition of subgroup
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A subgroup $H$ of a group $G$ is a subset of $G$ that is closed under the group operation. Denoted as $H\leq G$.
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#### Left and right cosets
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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 $a$.
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$$
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aH=\{x|a\sim x\}=\{x\in G|a^{-1}x\in H\}=\{x|x=ah\text{ for some }h\in H\}
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$$
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Similarly, $Ha$ is a right coset of $H$ for $a$.
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$$
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Ha=\{x|x\sim'a\}=\{x\in G|xa^{-1}\in H\}=\{x|x=ha\text{ for some }h\in H\}
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$$
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- Usually, the left coset and right cosets will give different partitions of $G$.
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- Use to prove lagrange theorem (partition of $G$ into cosets)
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#### Definition of normal subgroup
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A subgroup $H$ of a group $G$ is normal if $aH=Ha$ for all $a\in G$.
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### Isomorphism and homomorphism
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#### Definition of isomorphism
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Two groups $G$ and $G'$ are isomorphic if there exists a function $f:G\to G'$ such that
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- Homomorphism property is satisfied: $f(a*b)=f(a)f(b),\forall a,b\in G$
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- $f$ is injective: $f(a)=f(b)\implies a=b$
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- $f$ is surjective: $\forall a\in G',\exists b\in G$ such that $f(b)=a$
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#### Definition of homomorphism
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A homomorphism is a function that satisfies the homomorphism property.
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If $\phi:G\to G'$ is a homomorphism, then
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- $\phi(e)=e'$, where $e$ is the identity of $G$ and $e'$ is the identity of $G'$.
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- $\phi(a^{-1})=(\phi(a))^{-1}$ for all $a\in G$.
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- If $H\leq G$ is a subgroup, then $\phi(H)\leq G'$ is a subgroup.
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- If $K\leq G'$ is a subgroup, then $\phi^{-1}(K)\leq G$ is a subgroup.
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- $\phi$ is surjective if and only if $\operatorname{ker}(\phi)=\{e\}$ (the trivial subgroup of $G$).
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### Basic groups
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#### Trivial group
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The group $(\{e\},*)$ is called the trivial group.
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#### Abelian group
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A group $G$ is abelian if $a*b=b*a$ for all $a,b\in G$.
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- The smallest non-abelian group is $S_3$ (order 6).
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- Every abelian group is isomorphic to some direct product of cyclic groups of the form:
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$$
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\mathbb{Z}_{p_1^{n_1}}\times \mathbb{Z}_{p_2^{n_2}}\times \cdots \times \mathbb{Z}_{p_k^{n_k}}\times\underbrace{\mathbb{Z}\times \ldots \times \mathbb{Z}}_{m\text{ times}}
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$$
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#### Cyclic group
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A group $G$ is cyclic if $G$ is a subgroup generated by $a\in G$. (may be infinite)
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- The smallest non-cyclic group is Klein 4-group (order 4).
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- Every group with prime order is cyclic.
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- Every cyclic group is abelian.
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- If $G$ has order $n$, then $G$ is isomorphic to $(\mathbb{Z}_n,+)$.
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- If $G$ is infinite, then $G$ is isomorphic to $(\mathbb{Z},+)$.
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- If $G=\langle a\rangle$ and $H=\langle a^k\rangle$, then $|H|=\frac{|G|}{d}$ where $d=\operatorname{gcd}(|G|,|H|)$.
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- Every subgroup of cyclic group is also cyclic.
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#### Dihedral group
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The dihedral group $D_n$ is the group of all symmetries of a regular polygon with $n$ sides.
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- $|D_n|=2n$.
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- 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.
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#### Symmetric group
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The symmetric group $S_n$ is the group of all permutations of $n$ objects.
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- $S_n$ has order $n!$.
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- Every group $G$ is isomorphic to $S_A$ for some $A$.
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- Odd and even permutations
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- Every permutation can be written as a product of transpositions.
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- $A_n$ is the alternating group with order $\frac{n!}{2}$ consisting of all even permutations.
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- A non trivial homomorphism from $S_n$ to $(\Z_2,+)$ is given by $\sigma\mapsto \begin{cases} 0 & \text{if } \sigma\text{ is even} \\ 1 & \text{if } \sigma\text{ is odd} \end{cases}$ |