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

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

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

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 $a^n=e$ .

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

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

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)

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

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

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 $G$ ).

The group $(\{e\},*)$ is called the trivial 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: $\mathbb{Z}_{p_1^{n_1}}\times \mathbb{Z}_{p_2^{n_2}}\times \cdots \times \mathbb{Z}_{p_k^{n_k}}\times\underbrace{\mathbb{Z}\times \ldots \times \mathbb{Z}}_{m\text{ times}}$

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.

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.

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 $\sigma\mapsto \begin{cases} 0 & \text{if } \sigma\text{ is even} \\ 1 & \text{if } \sigma\text{ is odd} \end{cases}$