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Condition : Let $G \le S_n$ acts on set $[n]$ and the action of $G$ on $[n]$ is $A_n$ i.e. it induce an homomorphism $\phi : G \mapsto S_n$, where image$(\phi) = A_n$.

Prove or Disprove: If $G$ satisfy the above written condition then maximal subgroup of $G$ is unique ( there is one only ).

This how I am thinking : Every group action on cosets is block system (each coset is a block ), now for a minimal block system coset size should be small (that means maximal subgroup). The above defined group action gives unique minimal block system, so there has to be one maximal subgroup.

Any high level idea is also fine.

  • The only finite groups with a unique maximal subgroup are cyclic groups. See https://math.stackexchange.com/questions/39941 – Derek Holt Nov 19 '17 at 17:51
  • What is the point of even talking about group actions in this problem? The question is just if $A_n$ has a unique maximal subgroup. – anon Nov 19 '17 at 18:08
  • @ anon See above –  Nov 19 '17 at 18:11
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    This question makes very little sense in a number of ways. As anon says, the condition just says that $G=A_n$, so why introduce group actions, which are not relevant? Secondly, the only finite groups that have a unique maximal subgroup are cyclic groups, and $A_n$ is not cyclic for $n>3$, so the answer to the question is clearly no. Thirdly, the group $A_n$ acting on $n$ points is primitive, so it does not preserve any proper block systems, and hence your comments on block systems are not relevant. – Derek Holt Nov 20 '17 at 10:40

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Take $G = A_n$ and the conditions are satisfied. But $A_n$ doesn't have a unique maximal subgroup. In particular as a counterexample you can take $A_4$ and show that it has maximal subgroups of order $4$ and $3$.

Stefan4024
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  • Your counter example is very much valid, but do you have a any non-trivial example I mean when $G \le S_n$. I mean there are many $G$ which satisfy my condition in my question. –  Nov 19 '17 at 18:12
  • @sssss I don't really understand your request. I mean if you take any $G \le S_n$ then the homomorphism induced by the group action will have $G$ as it's image. In particular you have to have $G=A_n$ – Stefan4024 Nov 19 '17 at 18:14