### An Introduction to Galois Theory

This article only skims the surface of Galois theory and should probably be accessible to a 17 or 18 year old school student with a strong interest in mathematics.

### What's a Group?

Explore the properties of some groups such as: The set of all real numbers excluding -1 together with the operation x*y = xy + x + y. Find the identity and the inverse of the element x.

### Groups of Sets

The binary operation * for combining sets is defined as the union of two sets minus their intersection. Prove the set of all subsets of a set S together with the binary operation * forms a group.

# An Introduction to Mathematical Structure

##### Stage: 5

Published March 2005,April 2005,March 2011,April 2011,February 2011.

Here are the answers to the challenges:

2) Take the numbers 2, 4, 6, 8 and multiply them mod 10 (so just write down the last digit).
 x 2 4 6 8 2 4 8 2 6 4 8 6 4 2 6 2 4 6 8 8 6 2 8 4
3) Take the numbers 1, 2, 3, 4 and multiply them mod 5.
 x 1 2 3 4 1 1 2 3 4 2 2 4 1 3 3 3 1 4 2 4 4 3 2 1
4) Take the numbers 1, 3, 5, 7 and multiply them mod 8.
 x 1 3 5 7 1 1 3 5 7 3 3 1 7 5 5 5 7 1 3 7 7 5 3 1
5) Rotations of a square
 I Q Q² Q³ I I Q Q² Q³ Q Q Q² Q³ I Q² Q² Q³ I Q Q³ Q³ I Q Q²
6) Symmetries of a rectangle
 I V H R I I V H R V V I R H H H R I V R R H V I
7) The T-shirt!
 Same BTF I-O Both Same Same BTF I-O Both BTF BTF Same Both I-O I-O I-O Both Same BTF Both Both I-O BTF Same