Group

Definition

A group is defined as a set GG closed under a binary operation ★ and is usually instantiated as<G,><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. xyx★y is in GG, for all x,yx,y in GG

  2. Associativity

    1. (xy)z=x(yz)(x★y)★z = x★(y★z), for all x,y,zx,y,z in GG

  3. Identity

    1. There exists a single “ee” in GG such that xe=ex=xx★e = e★x = x

  4. Inverse

    1. There exists an x1x^{-1} in GG such that xx1=x1x=ex★x^{-1} = x^{-1}★x = e , where “ee” refers to the “ee” of the identity property

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

Examples

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

    1. closure ✅

    2. associativity ✅

    3. identity a+0=0+a=aa + 0 = 0 + a = a

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

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

    1. closure ✅

    2. associativity ✅

    3. identity a1=1a=aa1 = 1a = a

    4. inverse a1a=1aa=1a * \frac{1}{a} = \frac{1}{a} * a = 1, but 1a\frac{1}{a} does not necessarily exist in Z\mathbb{Z}

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