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How to find the number of subgroups of $S_4$ and list them?

For order $1$ it is {${e}$}, order $2$, we will consider $<(12)>$, $<(23)>$, $<(34)>$, $<(13)>$, $<(14)>$, $<(24)>$, for order $3$ we will consider $<(123)>$, $<(124)>$, $<(143)>$, $<(234)>$, for order 4 we will look at $<(1234)>$, $<(1324)>$, and so on. $A_4$ will also be there. I don't think these will be all ...

tattwamasi amrutam
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  • Why don't you do things in order? Of order two you have $;\langle (12)\rangle,,,\langle (13)\rangle,,,\langle (14)\rangle;$, etc. – DonAntonio Nov 14 '13 at 16:58
  • The subgroups that you have listed so far are each generated by one element, i.e. cyclic. Not all subgroups are cyclic. Inside of $S_4$, you should find an isomorphic copy of the Klein four group $C_2 \times C_2$, for instance. – Sammy Black Nov 14 '13 at 20:53
  • Also $<(1,2)(3,4)>$ is an example of a group of order $2$ that you did not list, and there groups of order $4$ that you did not list, such as $<(1,2),(3,4)>$. – DKal Nov 14 '13 at 20:56

2 Answers2

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An alternative approach is to start from the top and work down. For example, you know that $[S_4 : A_4 ]=2$, so your "first" subgroup is $S_4$. The subsequent subgroups would be $A_4$ and its conjugates. Now you localize the problem and just need to list the subgroups of $A_4$ and its conjugates.

Now $\vert A_4\vert =12$, so check if it has subgroups of index $2$ ($A_4$ has no subgroups of index 2, but what about $3$?). Continue the process. Eventually you get to subgroups of order $2$ and $3$, which are of course simple. Drawing a lattice of subgroups might help you avoid missing any subgroups.

JMag
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Using an old version of GAP can help you sorting these subgroups:

gap> LoadPackage("sonata");;
    s4:=SymmetricGroup(4);;
    e:=Subgroups(s4);
    Size(e);

 Group(()), Group([ (1,3)(2,4) ]), Group([ (1,4)(2,3) ]), 
 Group([ (1,2)(3,4) ]), Group([ (3,4) ]), Group([ (2,4) ]), Group([ (2,3) ]),
 Group([ (1,4) ]), Group([ (1,3) ]), Group([ (1,2) ]), Group([ (2,4,3) ]), 
 Group([ (1,3,2) ]), Group([ (1,3,4) ]), Group([ (1,4,2) ]), 
 Group([ (1,4)(2,3), (1,3)(2,4) ]), Group([ (1,2)(3,4), (3,4) ]), 
 Group([ (1,3)(2,4), (2,4) ]), Group([ (1,4)(2,3), (2,3) ]), 
 Group([ (1,2)(3,4), (1,3,2,4) ]), Group([ (1,3)(2,4), (1,2,3,4) ]), 
 Group([ (1,4)(2,3), (1,2,4,3) ]), Group([ (3,4), (2,4,3) ]), 
 Group([ (1,3), (1,3,2) ]), Group([ (1,3), (1,3,4) ]), 
 Group([ (1,4), (1,4,2) ]), Group([ (1,3)(2,4), (1,4)(2,3), (1,2) ]), 
 Group([ (1,2)(3,4), (1,4)(2,3), (1,3) ]), 
 Group([ (1,2)(3,4), (1,3)(2,4), (1,4) ]), 
 Group([ (1,3)(2,4), (1,4)(2,3), (2,4,3) ]), 
 Group([ (1,3)(2,4), (1,4)(2,3), (2,4,3), (1,2) ]) 

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In this updated but not a new version of GAP 4.6.4, you may use the following codes without calling sonata instead:

gap>s4:=SmallGroup(24,12);;
    e:=AllSubgroups(s4);
    Size(e);
Mikasa
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