Math4302 Modern Algebra (Lecture 3)

Groups

Properties for groups

• $(a*b)^{-1}=b^{-1}*a^{-1}$ (inverse)
• $a*b=a*c\implies b=c$ (cancellation on the left)
• $b*a=c*a\implies b=c$ (cancellation on the right)
• If $a*b=e$, then $b=a^{-1}$ (we can solve linear equations)

Additional notation

for $n\geq 1$,

• $a^n=a*a\cdot \cdots \cdot a$ (n times)
• $a^{-n}=a^{-1}\cdot \cdots \cdot a^{-1}$ (n times)

for $n=0$, $a^0=e$

We can easily prove this is equivalent to our usual sense for power notations.

That is

• $a^n*a^m=a^{n+m}$
• $(a^n)^m=a^{nm}$
• $a^{-n}=(a^{-1})^n$

Finite groups

Group with 4 elements.

Note $a,c$ are inverses and $b$ self inverse.

isomorphic to $(\mathbb{Z}_4,+)$, $(\{1,-1,i,-i\},\cdot)$

and we may also have

is

Cyclic groups

It is the group of integers modulo addition $n$.

• Associativity: $(a+b)+c=a+(b+c)$
• Identity: $a+0=a$
• Inverses: $a+(-a)=0$

For group with $4$ elements

Complex numbers

Consider $\{1,i,-1,-i\}$ with multiplication.

Note that if we replace $1$ with $0$ and $i$ with $1$, and $-1$ with $2$ and $-i$ with $3$, you get the exact the same table as $\mathbb{Z}_4$.