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  text: 'The collection of elements of the [-497] $S_n$ which are made by multiplying together an even number of permutations forms a subgroup of $S_n$.\n\nThis proves that the [-alternating_group] $A_n$ is well-defined, if it is given as "the subgroup of $S_n$ containing precisely that which is made by multiplying together an even number of transpositions".\n\n# Proof\n\nFirstly we must check that "I can only be made by multiplying together an even number of transpositions" is a well-defined notion; this [4hh is in fact true].\n\nWe must check the group axioms.\n\n- Identity: the identity is simply the product of no transpositions, and $0$ is even.\n- Associativity is inherited from $S_n$.\n- Closure: if we multiply together an even number of transpositions, and then a further even number of transpositions, we obtain an even number of transpositions.\n- Inverses: if $\\sigma$ is made of an even number of transpositions, say $\\tau_1 \\tau_2 \\dots \\tau_m$, then its inverse is $\\tau_m \\tau_{m-1} \\dots \\tau_1$, since a transposition is its own inverse.',
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