An Introduction to groups 1. Consider the set of four numbers 1,-1, i, -i. (a) What happens if you multiply any of these numbers by any of the others? ( Do you ever get any numbers out of the set?) (b) What do all four have in common? 2. Now consider the four rotations : Through 90°, 180°, 270°, 360° (a) What happens if you compose the transformations? (b) Could we pair up the rotations in this set with the numbers in question 1?
2. Would you say that the following axioms were true for the situations in 1 and 2. A set has four elements and these are combined with the binary operation * (in the first case * means multiply, in the second it means composition. Axiom 1. There is a member a of the set for which a*b = b for any b in the set (Identity) Axiom 2. For every member of the set b there is an element b-1 such that b*b-1=a (a as above) (Inverse) Axiom 3. if c and b are in the set so is b*c (Closure) Axiom 4. a*(b*c)=(a*b)*c ( Associative)
A set with a binary operation which satisfies these four axioms is called a GROUP.
3. Find a group with only two elements (order two).
4. Find a group with three elements (order three)
5. Consider the set of integers modulo 4. These are the remainders 0,1,2 and 3 which can be left after division by four. All integers are “represented” by these four numbers since we can set up an equivalence: 1 5 mod 4 because they leave the same remainder We can say that 3 11mod 4
18 2 mod 4 etc Fill in this table using addition mod 4 + 0 1 2 3 0 1 2 3
Fill in this table using the set in question 1. * 1 i -1 -i 1 i -1 -i How would you say the two groups compare?
Can you explain the reason for the same patterns among the elements?
Do you think all groups of order 4 have this same pattern amongst the elements? Draw up a table for the integers 1,2,3,4 modulo 5 with the operation “multiplication”.
Does this have the same structure?
HW ( to be turned in with the rest of this sheet) There are six symmetries of an equilateral triangle. What are they? Draw up the group table. Identify the Identity element.