Mathematics

Every Normal Subgroup Of An Contains A 3Cycle
Suppose H is normal in An and H is not {e}, and n>4 Then H contains a three cycle.
Assume an element not equal to {e} from H as x = y_{1}y_{2}...y_{s} Is a product of disjoint cycles y_{i} and I may assume that the length of the cycles decrease so that, length(y_{1}) >= length(y_{2}) >= length(y_{3}) >= .... >= length(y_{s})
I will write y_{1} = (a_{1},a_{2},a_{3},....,a_{m})
CASE 1) if m>3
put W = (a_{1},a_{2},a_{3})
then Wx(W^{1})(x^{1}) is in H. I.e.
(W)(y_{1}y_{2}...y_{s})(W^{1})(y_{s}^{1})(y_{s1}^{1})...(y_{2}^{1})(y_{1}^{1}) = (W)(y_{1}y_{2}...y_{s})(W^{1})(y_{1}^{1})(y_{2}^{1})...(y_{s}^{1})
(ie the y_{i} commute)
Now, W contains only symbols from y_{1} so W commutes with y_{2} to y_{s}
Wx(W^{1})(x^{1}) = Wy_{1}(W^{1})(y_{1}^{1}) = (a_{1},a_{2},a_{3})(a_{1},a_{2},a_{3},...,a_{m})(a_{3},a_{2},a_{1})(a_{m},...a_{2},a_{1}) = (a_{1},a_{2},a_{4})(a_{3}) = (a_{1},a_{2},a_{4}), and therefore H contains a three cycle. //
CASE 2) length(y_{1}) = length(y_{2}) = 3
write y_{1} = (a_{1},a_{2},a_{3})
y_{2} = (a_{4},a_{5},a_{6})
put W = (a_{2},a_{3},a_{4})
Then Wx(W^{1})(x^{1}) = (W)(y_{1})(y_{2})(W^{1})(y_{1}^{1})(y_{2}^{1}) as before
= (a_{2},a_{3},a_{4})(a_{1},a_{2},a_{3})(a_{4},a_{5},a_{6})(a_{4},a_{3},a_{2})(a_{3},a_{2},a_{1})(a_{6},a_{5},a_{4}) = (a_{1},a_{4},a_{2},a_{3},a_{5}) So H contains a cycle of length 5, so I reapply CASE 1 above, and therefore H contains a three cycle. //
CASE 3) length(y_{1}) = m=3 and length(y_{2}) = ... = length(y_{s}) = 2
then x^{2} = (y_{1}^{2})(y_{2}^{2})...(y_{s}^{2}) = y_{1}^{2}, which is also a three cycle in H. //
CASE 4) y_{i} for all i are transpositions
y_{1} = (a_{1},a_{2}), y_{2} = (a_{3},a_{4})
put W = (a_{2},a_{3},a_{4})
then Wx(W^{1})(x^{1}) = (a_{2},a_{3},a_{4})(y_{1}...y_{s})(a_{4},a_{3},a_{2})(y_{1}...y_{s}) = (a_{2},a_{3},a_{4})y_{1}y_{2}(a_{4},a_{3},a_{2})y_{1}y_{2}
(since y_{3}...y_{s} commute with (a_{4},a_{3},a_{2}) as these only appear in y_{1} and y_{2} and the cycles are ALL disjoint.)
Wx(W^{1})(x^{1}) = (a_{2},a_{3},a_{4})(a_{1},a_{2})(a_{3},a_{4})(a_{4},a_{3},a_{2})(a_{1},a_{2})(a_{3},a_{4}) = (a_{1},a_{4})(a_{2},a_{3}) which is in H
Now, I can take x = (a_{1},a_{4})(a_{2},a_{3}) and since n>4 I can choose a_{5} not equal to a_{1}, a_{2}, a_{3} or a_{4} and I may then put W=(a_{1},a_{4},a_{5}) and compute Wx(W^{1})(x^{1}) in H
Wx(W^{1})(x^{1}) = (a_{1},a_{4},a_{5})(a_{1},a_{4})(a_{2},a_{3})(a_{1},a_{5},a_{4})(a_{1},a_{4})(a_{2},a_{3}) = (a_{1},a_{5},a_{4})(a_{2})(a_{3}) = (a_{1},a_{5},a_{4}) which is a three cycle in H. //
Proof so far complete. //
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