Group and Field

Group

A group G, over a binary operation, $o$, and two elements $a, b \in G$, is defined with the following four axioms:

• Closure: A group is closed under $o$, that is, the result of the operation is also a member of that group.

$$a\ o\ b = c \in G$$

• Associativity: The result of a binary operation on three or more elements remains the same, regardless of the arrangement of parentheses.

$$(a\ o\ b)\ o\ c = a\ o\ (b\ o\ c), \forall\ c \in G$$

• Identity: There exists an identity element, $i$, under operation, $o$ such that operation between any element of the group, $c$, and $i$ results in $c$.

$$\exists\ i \in G\ |\ c\ o\ i = c, \forall\ c \in G$$

• Inverse: For each element, $c$, of the group, there exists an inverse element $\bar c$ such that operation between $c$ and $\bar c$ results in identity element, $i$.

$$\forall\ c \in G, \exists\ \bar c\ |\ c\ o\ \bar c = i$$

Note: In general, a scalar can be considered an element of a group and isn’t restricted to a real or complex number.

Abelian group

An abelian group is a group, $A$, under binary operation, $o$, with one additional axiom.

• Commutativity: The order of elements while applying a binary operation is irrelevant.

$$a\ o\ b = b\ o\ a, \forall\ a, b \in A$$

Field

A field, $F$, satisfies the following axioms:

• Additive group: All elements of $F$ form an abelian group under operation $+$, with $0$ as the additive identity.
• Multiplicative group: All elements of $F$, excluding $0$, form an abelian group under operation $\times$, with $1$ as the multiplicative identity.
• Distribution of $\times$ over $+$: Multiplying an element $a$ with the result of the addition of two elements $b$ and $c$ gives the same result as multiplying each element with $a$ first and then adding the results.

$$a\times (b+c)=(a\times b)+(a\times c), \forall\ a,b,c \in F$$

Examples

Below are some examples of fields.

Real numbers

The set of real numbers, $\R$, is a field with standard definitions of $+$ and $\times$.

• $\R$ is closed under $+$ and $\times$ because we’re adding or multiplying two real numbers results in a real number.
• Associativity holds in $\R$ for both $+$ and $\times$.
• Commutativity holds in $\R$ for both $+$ and $\times$.
• The numbers $0$ and $1$ are additive and multiplicative identities, respectively.
• The additive inverse of every element is the sign-flipped version of it, whereas the multiplicative inverse is its reciprocal.
• Distributivity of $\times$ over $+$ holds in $\R$.

Complex numbers

The set of complex numbers, $C$, and the set of rational numbers, $Q$, are fields with standard definitions of $+$ and $\times$.

Orthogonal matrices

A set of orthogonal matrices of given dimensions are fields. Here, the zero matrix is the additive identity, and the identity matrix is the multiplicative identity. Additive inverse is found by simply changing the signs of each entry. Finally, by definition, orthogonal matrices are always invertible.