Group Theory and Linear Algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 16 September 2014

Lecture 1: The clock and invertible elememnts

Number systems – 12, the clock

12= { 12 11 1 10 2 9 3 8 4 7 5 6 } 2+12 = 2 3+4 = 7 10+5 = 3 The product, or multiplication on 12 is given by m·n= m+m++m ntimes . For example 5·3=5+5+5=10+5=3.

The multiplication table for 12 is · 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5 6 7 8 9 10 11 12 2 2 4 6 8 10 12 2 4 6 8 10 12 3 3 6 9 12 3 6 9 12 3 6 9 12 4 4 8 12 4 8 12 4 8 12 4 8 12 5 5 10 3 8 1 6 11 4 9 2 7 12 6 6 12 6 12 6 12 6 12 6 12 6 12 7 7 2 9 4 11 6 1 8 3 10 5 12 8 8 4 12 8 4 12 8 4 12 8 4 12 9 9 6 3 12 9 6 3 12 9 6 3 12 10 10 8 6 4 2 12 10 8 6 4 2 12 11 11 10 9 8 7 6 5 4 3 2 1 12 12 12 12 12 12 12 12 12 12 12 12 12 12

Let x12. The element x is invertible if there exists y12 such that y·x=1. The inverse of 5 is 5, since 5·5=1. The inverse of 2 does not exist.

Let m>0. The invertible elements of m are x>0 such that

(a) 1xm,
(b) gcd(x,m)=1.

The invertible elements of 12 are 1,5,7,11.

The additive identity is 012 such that if x12 then 0+x=x and x+0=x. Note that 0=12 in 12.

Number systems – >0, the free monoid generated by 1.

>0= { 1,1+1,1+1+1, 1+1+1+1, } with addition given by concatenation. For example (1+1)+ (1+1+1) =1+1+1+1+1. An example of multiplication in >0 is (1+1+1+1)·x= x+x+x+x.

Let x>0. The set of multiples of x is x·>0= {x,x+x,x+x+x,}. Let a,b>0. The element b divides a, b|a, if ab>0.

Let a,b>0. The greatest common divisor of a and b, gcd(a,b), is the largest d>0 such that d|a and d|b.

The order on >0: Let a,b>0. Define a<b if there exists x>0 such that a+x=b.

A better definition of gcd(a,b) is:

Let a,b>0. The greatest common divisor of a and b, gcd(a,b), is d>0 such that

(a) d|a and d|b,
(b) If >0 and |a and |b then d.

Notes and References

These are a typed copy of Lecture 1 from a series of handwritten lecture notes for the class Group Theory and Linear Algebra given on July 26, 2011.

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