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 – $\raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$12\mathbb{Z}$}\right.,$ the clock

$$\raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$12\mathbb{Z}$}\right.=\left\{\begin{array}{ccccccc}& & & 12& & & \\ & & 11& & 1& & \\ & 10& & & & 2& \\ 9& & & & & & 3\\ & 8& & & & 4& \\ & & 7& & 5& & \\ & & & 6& & & \end{array}\right\}\begin{array}{ccc}{\displaystyle 2+12}& =& {\displaystyle 2}\\ {\displaystyle 3+4}& =& {\displaystyle 7}\\ {\displaystyle 10+5}& =& {\displaystyle 3}\end{array}$$ The product, or multiplication on $\raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$12\mathbb{Z}$}\right.$ is given by $$m·n=\underset{\underset{n\hspace{0.17em}\text{times}}{⏟}}{m+m+\cdots +m}\text{.}$$ For example $5·3=5+5+5=10+5=3\text{.}$

The multiplication table for $\raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$12\mathbb{Z}$}\right.$ is $$\begin{array}{ccccccccccccc}·& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12\\ 1& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12\\ 2& 2& 4& 6& 8& 10& 12& 2& 4& 6& 8& 10& 12\\ 3& 3& 6& 9& 12& 3& 6& 9& 12& 3& 6& 9& 12\\ 4& 4& 8& 12& 4& 8& 12& 4& 8& 12& 4& 8& 12\\ 5& 5& 10& 3& 8& 1& 6& 11& 4& 9& 2& 7& 12\\ 6& 6& 12& 6& 12& 6& 12& 6& 12& 6& 12& 6& 12\\ 7& 7& 2& 9& 4& 11& 6& 1& 8& 3& 10& 5& 12\\ 8& 8& 4& 12& 8& 4& 12& 8& 4& 12& 8& 4& 12\\ 9& 9& 6& 3& 12& 9& 6& 3& 12& 9& 6& 3& 12\\ 10& 10& 8& 6& 4& 2& 12& 10& 8& 6& 4& 2& 12\\ 11& 11& 10& 9& 8& 7& 6& 5& 4& 3& 2& 1& 12\\ 12& 12& 12& 12& 12& 12& 12& 12& 12& 12& 12& 12& 12\end{array}$$

Let $x\in \raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$12\mathbb{Z}$}\right.\text{.}$ The element $x$ is invertible if there exists $y\in \raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$12\mathbb{Z}$}\right.$ such that $$y·x=1\text{.}$$ The inverse of $5$ is $5,$ since $5·5=1\text{.}$ The inverse of $2$ does not exist.

Let $m\in {\mathbb{Z}}_{>0}\text{.}$ The invertible elements of $\raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$m\mathbb{Z}$}\right.$ are $x\in {\mathbb{Z}}_{>0}$ such that

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

The invertible elements of $\raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$12\mathbb{Z}$}\right.$ are $1,5,7,11\text{.}$

The additive identity is $0\in \raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$12\mathbb{Z}$}\right.$ such that if $x\in \raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$12\mathbb{Z}$}\right.$ then $0+x=x$ and $x+0=x\text{.}$ Note that $0=12$ in $\raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$12\mathbb{Z}$}\right.\text{.}$

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

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

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

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

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

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

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

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

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