Talk:Group theory

You may want to review Sets.

Definition of a Group

A group $(𝕊, *)$ is defined as any set $𝕊$ along with a binary operator $*$ such that the following is true:
(1) If $a ϵ 𝕊$ and $b ϵ 𝕊$ , then $(a * b) ϵ 𝕊$ .
(2) There exists an element $e ϵ 𝕊$ such that if $a ϵ 𝕊$ , then $e * a = a * e = a$ .
(3) If $a ϵ 𝕊$ , then there exists a $b ϵ 𝕊$ such that $a * b = b * a = e$ .
(4) If $a, b, c ϵ 𝕊$ , then $a * (b * c) = (a * b) * c$ .

It's quite interesting to note that this definition, while often the standard, holds some redundancies. For instance, replacing requirements 2 and 3 with the following "one-sided" definitions doesn't actually lessen the scope of the definitions:
(2b) There exists an element $e ϵ 𝕊$ such that if $a ϵ 𝕊$ , then $e * a = a$ .
(3b) If $a ϵ 𝕊$ , then there exists a $b ϵ 𝕊$ such that $b * a = e$ .

Before going forward with our study of groups, it's sometimes helpful to stop and put into English what the definition actually states:
(1) requires that the binary operator $*$ is closed under the set $𝕊$ , (2) requires the existence of an identity element, (3) requires the existence of inverses, and (4) allows the interchange of parentheses.

Exercises
1. Show that $(ℤ, +)$ (where + is the "usual" addition) is a group.
2. Show that the natural numbers with the usual addition is not a group.
3. Show that the set ${x ϵ ℝ : x > 0}$ with the usual multiplication is a group.
4. Show that $ℝ$ with the usual multiplication is not a group.
5. Is there a smallest number of elements a group can contain? If so, what is it?
6. Is there a greatest number of elements a group can contain? If so, what is it?
*7. What is the smallest set which contains the natural numbers and forms a group with the usual multiplication?
*8. Show that the two-sided definition of a group follows from the one-sided definition. (The proofs for the right-sided definition is similar to the left-sided one, so you may choose one or the other here - sorry, but not both)