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- Thread starter Poirot
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- Feb 15, 2012

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we know there is x,y,z in G with:

xHx

yKy

zMz

so the action is transitive. because we have an

x:H-->K

then xKx

otherwise xKx

now if xKx

since y takes K to M, either y takes H to K, in which case y corresponds to (1 2 3)

or y takes H to H, in which case y corresponds to (2 3).

in this last case, we have xy corresponds to (1 2)(2 3) = (1 2 3):

xyH(xy)

xyK(xy)

xyM(xy)

(and we didn't even need to use z).