Talk:Group theory

You may want to review Sets.

Definition of a Group

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

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 $e\epsilon \mathbb {S}$ such that if $a\epsilon \mathbb {S}$, then $e\mathbb {*} a=a$.
(3b) If $a\epsilon \mathbb {S}$, then there exists a $b\epsilon \mathbb {S}$ such that $b\mathbb {*} a=e$.

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 $\mathbb {*}$ is closed under the set $\mathbb {S}$, (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 $(\mathbb {Z} ,+)$(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 $\{x\epsilon \mathbb {R} :x>0\}$ with the usual multiplication is a group.
4. Show that $\mathbb {R}$ 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)