NoteNextra-origin/content/Math4302/Math4302_L3.md

Math4302 Modern Algebra (Lecture 3)

Groups

More examples for groups

Let \mathbb{Q}^+ be the set of positive rational numbers.

Then (\mathbb{Q}^+,\times) is a abelian group with identity 1 and inverse a^{-1}=\frac{1}{a}.

If we defined * by a*b=\frac{ab}{2}, then we have identity 2. a*e=\frac{ae}{2}=a, we have e=2.

and inverse a^{-1}a=\frac{a^2}{2}=2, therefore a^{-1}=\frac{4}{a}.

This is also an abelian group.

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.

*	e	a	b	c
e	e	a	b	c
a	a	b	c	e
b	b	c	e	a
c	c	e	a	b

Note a,c are inverses and b self inverse.

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

and we may also have

*	e	a	b	c
e	e	a	b	c
a	a	e	c	b
b	b	c	e	a
c	c	b	a	e

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

*	0	1	2	3
0	0	1	2	3
1	1	2	3	0
2	2	3	0	1
3	3	0	1	2

Complex numbers

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

*	1	i	-1	-i
1	1	i	-1	-i
i	i	-1	-i	1
-1	-1	-i	1	i
-i	-i	1	i	-1

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.