Let $G$ be finite group. Show that if $G$ is a solvable group, and derived length is $n$ then $G$ contains a abelian normal nontrivial subgroup $H$ with $G/H$ has derived length is $n-1$
$G$ contains a abelian normal nontrivial subgroup $H$ with $G/H$ has derived length is $n-1$
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abstract-algebra
group-theory
finite-groups
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0very nice, thank you about everything – 2012-10-07
1 Answers
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Hint: look to the last non-trivial element in the derived series, i.e.:
$G> G'>G''>...>G^{(n)} >1\,\,\,,\,\,\text{then take }\,\,\,H=G^{(n)}$