Proof: Proving Klein 4 Group is Not Isomorphic to ##Z_4##

[LagrangeEuler]

Homework Statement

Prove that Klein 4 group is not isomorphic with $Z_4$.

Homework Equations

Klein group has four elements $\{e,a,b,c\}$ such that $e^2=e,a^2=e,b^2=e,c^2=e$
As far as I know $Z_4$ group is $(\{\pm 1,\pm i\},\cdot)$. Right?

The Attempt at a Solution

As far as I know I can say group $Z_4$ is cyclic (all elements I could get as $i^n,n=1,2,3,4$) and group and Klein group is not.
Q.E.D.
Is this correct prove?
Klein group has four element of order $2$, and $Z_4$ group has one element of order $4$, two element of order $2$ and one element of order one. Right?

 

[Dick]

Homework Statement

Prove that Klein 4 group is not isomorphic with $Z_4$.

Homework Equations

Klein group has four elements $\{e,a,b,c\}$ such that $e^2=e,a^2=e,b^2=e,c^2=e$
As far as I know $Z_4$ group is $(\{\pm 1,\pm i\},\cdot)$. Right?

The Attempt at a Solution

As far as I know I can say group $Z_4$ is cyclic (all elements I could get as $i^n,n=1,2,3,4$) and group and Klein group is not.
Q.E.D.
Is this correct prove?
Klein group has four element of order $2$, and $Z_4$ group has one element of order $4$, two element of order $2$ and one element of order one. Right?

Yes, that's a good proof. But you've got some problems with counting orders. $e^1$ is also equal to $e$. Go back and count them carefully and say which elements have which orders.

 

[LagrangeEuler]

So $i^4=1=e$ has order $1$. $i^2=-1$ has order 2. $i^3=-i$ has order $4$ and $i$ has order $4$.

 

[Dick]

So $i^4=1=e$ has order $1$. $i^2=-1$ has order 2. $i^3=-i$ has order $4$ and $i$ has order $4$.

Yes, that's better. And the Klein group has 3 elements of order 2, and 1 element of order 1, yes?

 

[LagrangeEuler]

Yes! Thanks!