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text: 'The *alternating group* $A_n$ is defined as a certain subgroup of the [-497] $S_n$: namely, the collection of all elements which can be made by multiplying together an even number of [4cn transpositions].\nThis is a well-defined notion ([4hg proof]).\n\n%%%knows-requisite([4h6]):\n$A_n$ is a [-4h6] of $S_n$; it is the [quotient_group quotient] of $S_n$ by the [4hk sign homomorphism].\n%%%\n\n# Examples\n\n- A [49f cycle] of even length is an odd permutation in the sense that it can only be made by multiplying an odd number of transpositions.\nFor example, $(132)$ is equal to $(13)(23)$.\n- A cycle of odd length is an even permutation, in that it can only be made by multiplying an even number of transpositions.\nFor example, $(1354)$ is equal to $(54)(34)(14)$.\n\n- The alternating group $A_4$ consists precisely of twelve elements: the identity, $(12)(34)$, $(13)(24)$, $(14)(23)$, $(123)$, $(124)$, $(134)$, $(234)$, $(132)$, $(143)$, $(142)$, $(243)$.\n\n# Properties\n\n%%%knows-requisite([4h6]):\nThe alternating group $A_n$ is of [index_of_a_subgroup index] $2$ in $S_n$.\nTherefore $A_n$ is [4h6 normal] in $S_n$ ([4hl proof]).\nAlternatively we may give the homomorphism explicitly of which $A_n$ is the [49y kernel]: it is the [4hk sign homomorphism].\n%%%\n\n- $A_n$ is generated by its $3$-cycles. ([4hs Proof.])\n- $A_n$ is [4jc simple]. ([4jb Proof.])\n- The [4bj conjugacy classes] of $A_n$ are [alternating_group_five_conjugacy_classes easily characterised].',
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