Groups, Rings and Fields

• Groups, rings and fields endow sets with additional algebraic operations.
• A group is a set equipped with a single binary operation that exhibits certain properties akin to addition or multiplication.

A group $G$ is a set which is closed under an operation $\ast$: $\forall x,y\in G$, $x\ast y\in G$, and satisfies the following properties:

1. Identity: $\exists e\in G$ s.t. $\forall x\in G$, $x\ast e=x=e\ast x$, where $e$ is called the identity element.
2. Inverse: $\forall x\in G$, $\exists y\in G$ s.t. $x\ast y=e=y\ast x$, $e=$ identity element.
3. Associativity: multiplication is associative for $\forall x,y,z\in G$.

E.g. the set $\mathbb{Z}$ under $+$ is a group and the identity element is $0$.

Abelian Group $\Rightarrow$ multiplication is commutative $\forall x,y\in G$.

Popular symmetry Groups

• SO(N): all $N\times N$ orthogonal matrices with determinant $1$, representing rotations in $N$-dimensional space.
• SU(2): special unitary group of degree two, all $2\times 2$ matrices with determinant $1$. Important in the quantum realm — describe spins in / qubit states.
• SU(N): SU(2) extended to $N$ dimensions, helpful in the study of quantum entanglement.

Subgroups

A subgroup $H$ of a group $G$ is a subset of $G$ that is itself a group under the operation inherited from $G$.

Cosets and cyclic Groups

Cosets are used to partition a group into equivalence classes based on subgroups. Given a subgroup $H$ of a group $G$, the concept of cosets allows us to divide $G$ into distinct subsets.

Case: Given a group $G$ and a subgroup $H$ of $G$, the left coset of $H$, $g\in G$, is the set $gH=\{gh\mid h\in H\}$. The right coset is $Hg=\{hg\mid h\in H\}$.

Rings and Fields

A ring is a set $R$ equipped with $+$ and $\times$ operations s.t.

1. $(R,+)$ forms an abelian group.
2. Associativity of $\times$: $\forall a,b,c\in R$, $a\times (b\times c)=(a\times b)\times c$.
3. Distributive property: $\forall a,b,c\in R$,

$a\times (b+c)=(a\times b)+(a\times c)$

$(b+c)\times a=(b\times a)+(c\times a)$