query: Show that $A_n$ is the kernel of a group homomorphism of $S_n \rightarrow \{−1,1\}$.
positive_passages.docid: 45277
positive_passages.text: The alternating group is a normal subgroup of the symmetric group
negative_passages.docid: 98507
negative_passages.text: $\text{Aut}(F)$ is isomorphic to the multiplicative group of all $n\times n$ matrices over $\mathbb Z$
negative_passages.docid: 30327
negative_passages.text: $\ker \phi = (x_1-a_1, ..., x_n-a_n)$ for a ring homomorphism $\phi: R[x_1, ..., x_n] \to R$
negative_passages.docid: 90616
negative_passages.text: When is mapping $g$ to $g^{-1}$ a group homomorphism?
negative_passages.docid: 109236
negative_passages.text: Subgroups containing kernel of group morphism to an abelian group are normal.
negative_passages.docid: 88930
negative_passages.text: Homomorphism between free groups
negative_passages.docid: 13761
negative_passages.text: Automorphisms in $\mathbb{Z}_n$
negative_passages.docid: 104505
negative_passages.text: Is every normal subgroup the kernel of some endomorphism?
negative_passages.docid: 54911
negative_passages.text: The preimage of a normal subgroup under a group homomorphism is normal
negative_passages.docid: 39453
negative_passages.text: How do I prove that $S^n$ is homeomorphic to $S^m \Rightarrow m=n$?
negative_passages.docid: 121587
negative_passages.text: Let $H$ be a normal subgroup of index $n$ in a group $G$. Show that for all $g \in G, g^n \in H$
negative_passages.docid: 151699
negative_passages.text: Group of inner automorphisms of a group $G$
negative_passages.docid: 7938
negative_passages.text: Number of Homomorphisms from $\Bbb{Z}_m$ to $\Bbb{Z}_n$
negative_passages.docid: 32108
negative_passages.text: $ a\in \Bbb Z_n$ is invertible $\,\Rightarrow\gcd (a,n) =1$
negative_passages.docid: 7575
negative_passages.text: Show the commutator subgroup of $S_{n}$ is $A_{n}$ for $n \geq 5$
negative_passages.docid: 10998
negative_passages.text: Embedding $S_n$ in $A_{2n}$
negative_passages.docid: 27728
negative_passages.text: Question on a homomorphism of a set G.