# Abstract Algebra

## Group Theory

**Example:** What is the order of a given group?

**W|A Query:** `crystallographic group S6`

**Results:** W|A gives the order of the group (720), and tells us that it is nonabelian, nonsolvable, symmetric and transitive. However it seems like the number of groups W|A knows something about is limited at this time.

**Example:** How many groups are there of a given order?

**W|A Query:** `how many groups are there of order 1000`

**Results:** W|A first writes down the appropriate Mathematica command (FiniteGroupCount[1000]) and then gives the answer: 199.

**Implications:** So far W|A is not very capable of doing basic group theory. At this time students can find out about the various non-isomorphic groups of orders up to 20 (or sometimes higher) on the WEB simply by googling appropriately. But the power of Mathematica behind W|A will allow them to go beyond small numbers.

# Analysis

## Special subsets of the Real Line

**Example:** What does W|A know about the Cantor Set?

**W|A Query:** `cantor set`

**Results:** W|A opens a box and puts in it a small number for iterations (5) and then draws the partial Cantor set drawn after five iterations of the defining recursion. We have the option to modify the number of iterations and see how this changes the result. At each stage we are given information about the length of the set constructed.