I've been trying to prove something that seems obvious but have had no success thus far:(adsbygoogle = window.adsbygoogle || []).push({});

say G is a finite group and H and K are proper subgroups, if K contains a conjugate of H, then it isn't possible to have G=HK.

Proof anybody? I'm happy if one can prove the special case below:

It's fairly easy to show one can't have the product of a proper subgroup and it's conjugate equal to a group i.e. it isn't possible that Hx^{-1}Hx=G unless H=G. Can you show it for an arbitrary product of conjugates? i.e. if [tex]\Pi_{x \in G} H^x=G[/tex] then H=G. Note that [tex]H^x=x^{-1}Hx[/tex]

thanks for the help

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# Finite groups

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