# Math Help - Klein four-group is normal group of A4

1. ## Klein four-group is normal group of A4

I have shown that in strait forward way .
gVg^(-1) = V

But is there some other way to show that ?

Sylow dint worked for me here .

Thanks for help .

2. Originally Posted by s.lateralus
I have shown that in strait forward way .
gVg^(-1) = V

But is there some other way to show that ?

Sylow dint worked for me here .

Thanks for help .
What you have done looks like what I would have suggested!

3. some other ideas ?

4. Originally Posted by s.lateralus
some other ideas ?

If you already know that two permutations in $S_n$ are conjugate iff they have the very same cycle decomposition, then it is straightforward that $V$ is a normal subgroup of $S_4$ and, thus, also of $A_4$ since $V\subset A_4$ .

Tonio

5. Originally Posted by tonio
If you already know that two permutations in $S_n$ are conjugate iff they have the very same cycle decomposition, then it is straightforward that $V$ is a normal subgroup of $S_4$ and, thus, also of $A_4$ since $V\subset A_4$ .

Tonio
some example will really help.

6. Originally Posted by s.lateralus
Well, for example $(12)(345), (14)(235)$ are conjugate permutations in $S_n\,,\,n\ge 5$ , since they both are the product of a 2-cycle and a 3-cycle, but none of them is conjugate to $(13)(24)$ since this last is the product of two 2-cycles (transpositions), and etc.
8. Also, Klein is the unique subgroup of $A_4$ such as $|H|=4$, so it's normal.