Does a homomorphic image of even permutations consist of even permutations?

abstract-algebra group-theory permutations

  • $A_n$ is the set of products of an even number of transpositions

    this is one common definition of $A_n$

  • $A_n$ is the set of products of products of two transpositions

    just group pairs of transpositions

  • $A_n$ is the set of products of 3-cycles

    $(12)(23)=(123)$, $(12)(34)=(123)(234)$

  • $A_n$ is the set of products of squares

    $(123)=(132)^2$ and the square of every permutation is even

  • $f$ maps products of squares to products of squares

    $f$ is a homomorphism

  • $f$ maps $A_n$ into $A_n$.

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