1. ## homomorphisms and isomorphisms

I need help on this question!

Suppose G is an Abelian Group and let f:G--> G be the function defined by f(x)=x^3

i) show that f is a group homomorphism
ii) show that f is an group homomorphism iff lGl is not divisible by 3

2. Originally Posted by akvirdee
I need help on this question!

Suppose G is an Abelian Group and let f:G--> G be the function defined by f(x)=x^3

i) show that f is a group homomorphism
ii) show that f is an group homomorphism iff lGl is not divisible by 3

You must show $\displaystyle (ab)^3=a^3b^3\,,\,\,\forall\,a,b\in G$ . in (ii) I think it must be "isomorphism", and then you must show $\displaystyle a^3=1\Longleftrightarrow a=1$ . Try it and write back if you get stuck somewhere.

Tonio

3. thankyou for your help..the 1st part is pretty straight forward, however I did make a mistake with the second part it states

ii) show that f is an isomorphism iff lGl is not divisible by 3.

I don't quite understand how showing a^3=1 => a=1 shows that lGl is not divisible by 3.

4. Originally Posted by akvirdee
thankyou for your help..the 1st part is pretty straight forward, however I did make a mistake with the second part it states

ii) show that f is an isomorphism iff lGl is not divisible by 3.

I don't quite understand how showing a^3=1 => a=1 shows that lGl is not divisible by 3.
If it is true that for all $\displaystyle a\in G\,,\,\,a^3=1\Longrightarrow a=1$ , then the group has no element of order 3 and thus its order cannot be divisible by 3 (read about Cauchy Theorem).

Tonio

5. Originally Posted by tonio
If it is true that for all $\displaystyle a\in G\,,\,\,a^3=1\Longrightarrow a=1$ , then the group has no element of order 3 and thus its order cannot be divisible by 3 (read about Cauchy Theorem).

Tonio