Show that A4 has no subgroups of order 6SolutionShow that A4

Show that A4 has no subgroups of order 6

Solution

Show that A4 has no subgroup of order 6.

Note, |A4|=4!/2=12

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Suppose A4>H,|H|=6.
Then |A4/H|=[A4:H]=2.
So HA4 so consider the homomorphism
:A4A4/H
let xA4 with |x|=3 (i.e. in a 3-cycle)
then 3 divides |(x)|
so as |A4/H|=2 we have |(x)| divides 2
so (x)=eH so xH
so H contains all 3-cycles
but A4 has 8 3-cycles
8>6, A4 has no subgroup of order 6.