# Thread: 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
some example will really help.

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.
Of course, one must know first that ANY permutation is expressable as the product of DISJOINT cycles...

Tonio

7. If you want a geometric interpretation, use the fact that A4 is the group of rotations of a regular tetrahedron. The tetrahedron has three axes, namely the lines joining the midpoints of opposite edges. A rotation of the tetrahedron takes axes to axes, and thus induces a homomorphism from A4 to S3, whose kernel is the Klein group (consisting of the identity map together with rotations through 180º about each axis).

8. Also, Klein is the unique subgroup of $A_4$ such as $|H|=4$, so it's normal.