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いつも大変お世話になっております。

If G is finite and if #G/#H is a prime number, what can be said about
the subgroups X with H⊂X⊂G?
という問題です。

#G/#H=p(:素数)とすると
#G=pなら,#H=1だからX={e}(:単位群)かX=Gがあり得る。
#G=p^2なら#H=pでX=H(=Z_p)かX=G.
#G=pq (qは素数)なら#H=qだからp<qならX=HかX=G、p>qならX=Z_pでH⊂X⊂Gは言えませんよね?
#G=pn (nは合成数)なら#H=nでgcd(p,n)=1とgcd(p,n)≠1では何と言えますでしょうか?


吉田京子