Definition of a group

Question

What defines a group mathematically, please explain both in Mathematical language and in English if possible.

My current understanding: Four things are required to define a group:

Closure - Any binary operation completed upon two elements of a group, must always equal a third element that is already contained within the group. So for all a,b element of G, g*h element of G.

Associativity - If completing a binary operation between three different elements of a group, the order is irrelevant.

Identity element - There must exist some element a in G such that a*b=b(so 1 in the real numbers for multiplication, or 0 for addition)

Inverse element - There must be inverse element such that $b*b^{-1}$ = e

Answer 1

A group is a collection of objects, along with some defined binary operation, that meet the following four criterion:

1. The group exhibits closure under the operation. That is to say, when the operation is performed on any two elements in the group, the result is one of the elements in the group. For example, if the operation were addition, and I take two numbers from the collection, their sum is also a member of the group.

2. The group exhibits associativity under the group operation which we will call $*$. That is to say, given any three arbitrary elements in the group $a, b, c$, $(a*b)*c=a*(b*c)$.

3. There is a unique identity for every element in the group. That is to say, there exists one unique identity element $e$, such that for any arbitrary element $a$ in the group, $a*e=e*a=a$.

4. For any arbitrary element $a$ in the group, there exists a unique element $b$ in the group, such that $a*b=b*a=e$. That is to say that every element has one inverse, and when the operation is performed on the two elements, the result is the identity.

If these four properties hold, then we can say that this collection of elements forms a group.

The set of all integers $\mathbb{Z}=\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$ forms a group under the operation of simple addition. We can write this group in compact form as $<\mathbb{Z}, +>$. Is this statement true? Consider our four criterion for a group:

1. The group exhibits closure. If any two integers are added together, no matter which element they may be, the result will also be an integer.

2. Addition is associative. The order by which one chooses to add integers does not impact the result. for any integers $a, b, c$, $(a + b) + c = a + (b + c)$.

3. There is an identity element $e$, and it is the number $0$. Given any arbitrary element $a$ in the group, including the identity element, $a + 0 = a$.

4. Each element $a$ has its own unique inverse element $a^{-1}$, and $a + a^{-1} = e$. That element is the negation of $a$. For example, $2 + (-2) = 0$, or $(-3) + 3 = 0$. The identity is it's own inverse; $0 + 0 = 0$.

Since the four criterion are met, $<\mathbb{Z}, +>$ forms a group.

Welcome to groups. :))