Two subgroups are conjugate

Asked Aug 19 '20 at 22:25

Active Aug 19 '20 at 22:25

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Let $G$ a un group and $n\geq 2$ be an integer. Let $H_{1}$ and $H_{2}$ be two subgroups of $G$ that satisfy

$[G:H_{1}]=[G:H_{2}]=n \hspace{0.4cm} $ and $\hspace{0.4cm} [G:(H_{1}\cap H_{2})]=n(n-1).$

Prove that $H_{1}$ and $H_{2}$ are conjugate in $G$.

I don't know how can i start, because i know that $G/H_{1}$ and $G/H_{2}$ have the same number of elements and for $n=2$ $H_{1}$ and $H_{2}$ are normal subgroups of $G$ and maybe i have that.But for $n>2$?How can i find some element in $G$ to show the conjugation? Can you give me some hint to prove this proposition? Thank you.

asked Aug 19 '20 at 22:25

angie duque

1

There's an answer, and hints, and a long discussion there, so you can choose what you want. – David A. Craven Aug 19 '20 at 22:32
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This is Problem 7 from the IMC 2020 https://www.imc-math.org.uk/?year=2020§ion=problems&item=prob7q – Alexdanut Aug 19 '20 at 22:56
yes, that is the question.Thank you – angie duque Aug 19 '20 at 22:58

Two subgroups are conjugate

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