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)