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Prove that if H and K are two solvable normal subgroup of G then HK is also solvable normal subgroup of G.

I proved that every (H i K i ) is a normal subgroup of (H i+1 K i+1 ) but now I am struck in proving that Factors of Normal series of HK are abelian

Nandini
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  • Here's a MathJax tutorial :) – Shaun Nov 22 '24 at 20:59
  • Duplicate of https://math.stackexchange.com/questions/39814/for-g-a-group-and-h-unlhd-g-then-g-is-solvable-iff-h-and-g-h-are-solv and others that can be found by using site search within one minute. – Martin Brandenburg Nov 22 '24 at 20:59
  • And also here, and here. – Dietrich Burde Nov 22 '24 at 21:04
  • While equivalent, the first linked question doesn't immediately appear to be the same, so I can understand OPs missing that.

    I don't see a solution to the second, so while it is a duplicate, it doesn't help OP.

    I agree we should avoid duplicates.

     – NewViewsMath Nov 22 '24 at 21:07
  • This question is a popular one, so it is not surprising, that it has been answered often. Sometimes one has to read the answers, to see this. Here the "key" is to see that $AB/B\cong A/(A\cap B)$ is solvable, and $B$ is solvable, hence $AB$ is solvable. This is the argument in many answers (see the duplicates). – Dietrich Burde Nov 23 '24 at 21:07

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