# Calculating Galois Groups....

• Mar 14th 2010, 06:18 PM
Piman
Calculating Galois Groups....
I am currently going through Galois Theory by Ian Stewart. I am on Chapter 22 of the third edition and trying to solve exercise 22.6 at the end of the chapter which is the following:

Show that any transitive group of S4 is one of the following S4, A4, D8, V or Z4, defined as follows:

A4 = alternating group of degree 4.
V= {1,(12)(34),(13)(24),(14)(23)}.
D8= group generated by V and (12).
Z4 = group generated by (1234).

Amy ideas?

Thanks
• Mar 14th 2010, 08:13 PM
tonio
Quote:

Originally Posted by Piman
I am currently going through Galois Theory by Ian Stewart. I am on Chapter 22 of the third edition and trying to solve exercise 22.6 at the end of the chapter which is the following:

Show that any transitive group of S4 is one of the following S4, A4, D8, V or Z4, defined as follows:

A4 = alternating group of degree 4.
V= {1,(12)(34),(13)(24),(14)(23)}.
D8= group generated by V and (12).
Z4 = group generated by (1234).

Amy ideas?

Thanks

Yes: $S_4$ has ony 24 elements so you can write them all down and check by yourself that indeed the above subgroups are transitive and that if any other subgroup is transitive then it has to be one of the above. Sylow theory can come handy here (for example, to show that any sbgp. of order 8 is isomorphic with $D_8$ ...).

Tonio
• Mar 18th 2010, 10:19 AM
Piman
Almost there...
Tonio,

I have managed to find the 24 elements of S4 and have successfully checked that the subgroups are transitive using very basic methods.

However, I am now wondering how I can show that these are the only transitive subgroups of S4 ?

Also, with regard to your mention of Sylow, how can this be used effectively to solve the problem, so I have an alternative way to present the solution.

Cheers,