Register to reply

Any group of 3 elements is isomorphic to Z3

by kathrynag
Tags: elements, isomorphic
Share this thread:
kathrynag
#1
Nov1-10, 07:40 PM
P: 607
1. The problem statement, all variables and given/known data
Prove that any group with three elements is isomorphic to [tex]Z_{3}[/tex]


2. Relevant equations



3. The attempt at a solution
Let G be the group of three elements
We have an isomorphism if given c:G--->[tex]Z_{3}[/tex],
if c is one-to -one and onto and c(ab)=c(a)c(b)

First, we check one-to-one
We want c(a)=c(b) to imply a=b
My problem here is how to define c(a), c(b).
Onto:
We want c(a)=x and want to solve for a?
c(ab):
Same problem with not knowing what c(ab) is
Phys.Org News Partner Science news on Phys.org
New model helps explain how provisions promote or reduce wildlife disease
Stress can make hard-working mongooses less likely to help in the future
Grammatical habits in written English reveal linguistic features of non-native speakers' languages
micromass
#2
Nov1-10, 07:45 PM
Mentor
micromass's Avatar
P: 18,019
Take a group of three elements {e,a,b}. Since the order of every element must be three, we have that b=aČ. Thus the group is {e,a,aČ}.

Define the map G --> Z3 by
e ---> 0
a ---> 1
b ---> 2

It is easily checked that this is indeed an iso.
kathrynag
#3
Nov1-10, 07:50 PM
P: 607
So c(a)=c(b)
c(a)=1
c(b)=2
1=2 not true, but that means it's not an isomorphism

micromass
#4
Nov1-10, 07:53 PM
Mentor
micromass's Avatar
P: 18,019
Any group of 3 elements is isomorphic to Z3

Uh what? c(a) doesnt equal c(b)??? does it?
kathrynag
#5
Nov1-10, 07:59 PM
P: 607
I thought for the one to one part, you assume c(a)=c(b)
micromass
#6
Nov1-10, 08:02 PM
Mentor
micromass's Avatar
P: 18,019
Yes... never mind...
kathrynag
#7
Nov1-10, 08:06 PM
P: 607
but I assumed that but that amounts to 1=2. How does that work?
micromass
#8
Nov1-10, 08:15 PM
Mentor
micromass's Avatar
P: 18,019
You assumed that c(a)=c(b), and from that assimption followed that 1=2. So your assumption is wrong, and thus [tex]c(a)\neq c(b) [/tex]
kathrynag
#9
Nov1-10, 08:33 PM
P: 607
Then I guess I don't see how to show 1-1

onto
y=c(x)
Do I just take any element, say a
y=c(a)=1
y=1, but we want to solve for x I thought

c(a)c(b)
1*2
2=c(ab)
micromass
#10
Nov1-10, 08:35 PM
Mentor
micromass's Avatar
P: 18,019
You show 1-1 by a simple proof by contradiction. I'm sorry, but a math major really should be able to do such a thing...


Register to reply

Related Discussions
Abstract Alg- Group theory and isomorphic sets. Calculus & Beyond Homework 3
Proving a group G is isomorphic to D_10 Calculus & Beyond Homework 1
Showing Subgroups of a Permutation Group are Isomorphic Calculus & Beyond Homework 2
Generation of isomorphic fields by separate algebraic elements Linear & Abstract Algebra 13
Group VI elements Biology, Chemistry & Other Homework 1