query: Asymptotics of the number distinct of subgroups of $S_n$ as $n$ goes to infinity
positive_passages.docid: 15396
positive_passages.text: Enumerating all subgroups of the symmetric group
negative_passages.docid: 146098
negative_passages.text: Number of nonempty subsets of a set with $n$ elements
negative_passages.docid: 67554
negative_passages.text: In a group, the number of non-self-inverse elements is even
negative_passages.docid: 37012
negative_passages.text: The number of summands $\phi(n)$
negative_passages.docid: 15000
negative_passages.text: Show that number of elements in $(Z/nZ)^*$ is $\phi (n)$
negative_passages.docid: 96680
negative_passages.text: Generators of symmetric group $S_n$
negative_passages.docid: 128834
negative_passages.text: Proving the number of even and odd permutations of a subgroup $H<S_{n}$ are equal, provided $H$ is not contained in $A_{n}$
negative_passages.docid: 84132
negative_passages.text: Number of preimages of a group element under a homomorphism.
negative_passages.docid: 133770
negative_passages.text: Subgroups of finite abelian groups.
negative_passages.docid: 108691
negative_passages.text: Characterize the natural numbers $n$ such that there is a surjective group homomorphism from $S_n$ to $S_{n-1}$.
negative_passages.docid: 58863
negative_passages.text: Proof that $A_n$ the only subgroup of $ S_n$ index $2$.
negative_passages.docid: 71262
negative_passages.text: Subgroups of order $9$ isomorphic to $C_3 \times C_3$ in $S_6$
negative_passages.docid: 33945
negative_passages.text: Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups
negative_passages.docid: 124224
negative_passages.text: How many groups of order $n$ with center {e} exist?
negative_passages.docid: 41231
negative_passages.text: Subgroups of (Z,+) of order n
negative_passages.docid: 96614
negative_passages.text: Finite number of subgroups $\Rightarrow$  finite group
negative_passages.docid: 50762
negative_passages.text: If $S_n$ has cyclic subgroups of these orders or not?