Group

Definition

A group is defined as a set $G$ closed under a binary operation ★ and is usually instantiated as$<G, ★>$.

If our operation ★ = $+$ or is addition, we call this an Additive Group.

If our operation ★ = $*$ or is multiplication, we call this a Multiplicative Group.

Properties

Groups hold 4 properties:

1. Closure aka “closed”

   1. $x★y$ is in $G$, for all $x,y$ in $G$

2. Associativity

   1. $(x★y)★z = x★(y★z)$, for all $x,y,z$ in $G$

3. Identity

   1. There exists a single “$e$” in $G$ such that $x★e = e★x = x$

4. Inverse

   1. There exists an $x^{-1}$ in $G$ such that $x★x^{-1} = x^{-1}★x = e$ , where “$e$” refers to the “$e$” of the identity property

If the Commutative property is also valid ($x★y = y★x,$ for all $x, y$ in $G$), then the group is called an “Abelian Group.”

Examples

1. $<\mathbb{Z}$, $+>$ additive for the set of all integers ✅ (valid group)

   1. closure ✅

   2. associativity ✅

   3. identity $a + 0 = 0 + a = a$ ✅

   4. inverse $a + (-a) = (-a) + a = 0$ ✅

2. $<\mathbb{Z}$, $*>$ multiplicative for the set of all integers ❌ (not a valid group)

   1. closure ✅

   2. associativity ✅

   3. identity $a1 = 1a = a$ ✅

   4. inverse $a * \frac{1}{a} = \frac{1}{a} * a = 1$, but $\frac{1}{a}$ does not necessarily exist in $\mathbb{Z}$ ❌