5

There is a method of calculating all the subgroups of dihedral group $D_{2n}$.I want to know that is there any method to find all the subgroups of $S_{4}, A_{4} $. I have found all the subgroups of both $S_{4}$ and $A_{4}$ somewher on google but want to know how to find them. Secondly I want to know that how can we say that our lattice diagram is true? And why it is called lattice of subgroups? Thanks in advance

Prince Khan
  • 1,564

1 Answers1

1

For $S_4$ and $A_4$, the groups are small enough that one can find all their subgroups by hand. For a general symmetric group $S_n$, as far as I know, there is no method to find all its subgroups (although the O'Nan-Scott theorem gives information on the maximal subgroups of symmetric groups).

As for why we call it the lattice of subgroups, it is because subgroups, together with inclusion, form a lattice.

Pierre-Guy Plamondon
  • 11,138
  • 2
    yes $S_ {4} $ and $A_{4 } $ are small thatswhy I am interested in them. I had found all their subgroups and calculated them by hand too. But $S_{4} $ is not yet too small and to find its non-cyclic subgroups is bit tough too. I had calculated all the subgroups of $S_{4} $ which were 30. Now I want to know that is there a method to verify that it has 30 subgroups or we have to check the remaining possibilities by hand? – Prince Khan Dec 10 '16 at 18:45
  • 1
    Again, as far as I know, there is no general method for $S_4$, other than brute force. – Pierre-Guy Plamondon Dec 10 '16 at 18:47
  • 2
    Fine! Let me ask one more question please. If $x,y \in G $, then $ \cup \subseteq <x,y> $. Am I right? If yes, than what should be added to the left hand side so that we could get the equality. (I am sure this would save our time while finding all the subgroups of $S_{4} $) – Prince Khan Dec 10 '16 at 19:04
  • You are basically asking "What is the smallest subgroup containing $x$ and $y$", and this is not a question that has an easy answer in general. – Pierre-Guy Plamondon Dec 10 '16 at 19:06
  • 2
    No! the smallest subgroup containing $x$ and $y$ is basically $<x,y> $. I am interested to know whether there is a relation between the subgroup genreted by $x,y $ and the subgroups genereted by each of $x $ and $y $ – Prince Khan Dec 10 '16 at 19:10
  • 1
    I'm trying to say that there is no easy answer. An example which may convince you: $S_n = \langle (12), (12\ldots n) \rangle$, but the relationship between $\langle (12) \rangle$, $\langle (12\ldots n) \rangle$ and the whole of $S_n$ seems rather complicated. – Pierre-Guy Plamondon Dec 10 '16 at 19:16
  • 1
    Thank You so much Sir. I will not take your much time. Its my last question. Is $ \exists $ any such relation in the groups $ \mathbb Z_{n} $ and $ \mathbb Z $ – Prince Khan Dec 10 '16 at 19:26
  • 1
    See this link for the finite case, and this one for $\mathbb{Z}$. – Pierre-Guy Plamondon Dec 10 '16 at 19:37