query: Intuitive explanation for why the group $S_n$ has $n!$ elements
positive_passages.docid: 150156
positive_passages.text: Proving that $S_n$ has order $n!$
negative_passages.docid: 7575
negative_passages.text: Show the commutator subgroup of $S_{n}$ is $A_{n}$ for $n \geq 5$
negative_passages.docid: 150854
negative_passages.text: Identifying a quotient group (NBHM-$2014$)
negative_passages.docid: 156319
negative_passages.text: Show that a group with $|G| = 33$ contains an element of order $3$
negative_passages.docid: 144732
negative_passages.text: If $[G:H]=n$, then $g^{n!}\in H$ for all $g\in G$.
negative_passages.docid: 130081
negative_passages.text: Show that $A_n$ is the kernel of a group homomorphism of $S_n \rightarrow \{−1,1\}$.
negative_passages.docid: 124224
negative_passages.text: How many groups of order $n$ with center {e} exist?
negative_passages.docid: 131519
negative_passages.text: intuition on the fundamental group of $S^1$
negative_passages.docid: 33216
negative_passages.text: Prove that the order of an element in $S_n$ equals the least common multiple of the lengths of the cycles in its cycle decomposition.
negative_passages.docid: 123560
negative_passages.text: Intuition behind why order of a subgroup divides order of a group.
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: 64527
negative_passages.text: Show that an operation of elements in a finite group contains each element exactly once
negative_passages.docid: 68201
negative_passages.text: Cyclic group of order $n$
negative_passages.docid: 29717
negative_passages.text: Not sure how to go about this group theory proof
negative_passages.docid: 52089
negative_passages.text: Group of $2n$ elements, $n$ odd, is not simple
negative_passages.docid: 137691
negative_passages.text: What are the self-normalizing subgroups of $S_n$?
negative_passages.docid: 933
negative_passages.text: How many groups exist with order $n$ (two isomorphic groups are treated as the same group)