## Group Theory and Linear Algebra

Last updated: 16 September 2014

## Lecture 1: The clock and invertible elememnts

### Number systems – $ℤ}{12ℤ},$ the clock

$ℤ12ℤ= { 12 11 1 10 2 9 3 8 4 7 5 6 } 2+12 = 2 3+4 = 7 10+5 = 3$ The product, or multiplication on $ℤ}{12ℤ}$ is given by $m·n= m+m+⋯+m ⏟n times .$ For example $5·3=5+5+5=10+5=3\text{.}$

The multiplication table for $ℤ}{12ℤ}$ 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 $x\in ℤ}{12ℤ}\text{.}$ The element $x$ is invertible if there exists $y\in ℤ}{12ℤ}$ such that $y·x=1.$ The inverse of $5$ is $5,$ since $5·5=1\text{.}$ The inverse of $2$ does not exist.

Let $m\in {ℤ}_{>0}\text{.}$ The invertible elements of $ℤ}{mℤ}$ are $x\in {ℤ}_{>0}$ such that

 (a) $1\le x\le m,$ (b) $\text{gcd}\left(x,m\right)=1\text{.}$

The invertible elements of $ℤ}{12ℤ}$ are $1,5,7,11\text{.}$

The additive identity is $0\in ℤ}{12ℤ}$ such that if $x\in ℤ}{12ℤ}$ then $0+x=x$ and $x+0=x\text{.}$ Note that $0=12$ in $ℤ}{12ℤ}\text{.}$

### Number systems – ${ℤ}_{>0},$ the free monoid generated by $1\text{.}$

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

Let $x\in {ℤ}_{>0}\text{.}$ The set of multiples of $x$ is $x·ℤ>0= {x,x+x,x+x+x,…}.$ Let $a,b\in {ℤ}_{>0}\text{.}$ The element $b$ divides $a,$ $b|a,$ if $a∈bℤ>0.$

Let $a,b\in {ℤ}_{>0}\text{.}$ The greatest common divisor of $a$ and $b,$ $\text{gcd}\left(a,b\right),$ is the largest $d\in {ℤ}_{>0}$ such that $d|a$ and $d|b\text{.}$

The order on ${ℤ}_{>0}\text{:}$ Let $a,b\in {ℤ}_{>0}\text{.}$ Define $a if there exists $x\in {ℤ}_{>0}$ such that $a+x=b\text{.}$

A better definition of $\text{gcd}\left(a,b\right)$ is:

Let $a,b\in {ℤ}_{>0}\text{.}$ The greatest common divisor of $a$ and $b,$ $\text{gcd}\left(a,b\right),$ is $d\in {ℤ}_{>0}$ such that

 (a) $d|a$ and $d|b,$ (b) If $\ell \in {ℤ}_{>0}$ and $\ell |a$ and $\ell |b$ then $\ell \le d\text{.}$

## 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.