Topic talk:Group theory

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You may want to review Sets.

[edit] 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)

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