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  "body": "\n## Example\n\nIn the [[symmetric group|0515A6D2Y1]] $S_3$, let $A_3 = \\{e, (1\\,2\\,3), (1\\,3\\,2)\\}$. This is the [[alternating group|0Z5905Q73D]] of degree $3$.\n$A_3$ has $3$ elements and $S_3$ has $6$, so $A_3$ has [[index|04QEBF8S49]] $2$.\nThe two left cosets are $A_3$ and the set of the three transpositions. The two right cosets are the same two sets.\nThus $gA_3 = A_3g$ for each $g \\in S_3$, and $A_3$ is normal in $S_3$.\n\n## Remarks\n\nThis shows the result that [[subgroups of index two are normal|0V547YQRMV]].\n",
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