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October 21st, 2017, 08:42 AM   #1
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Residually solvable group

I need help to answer the following problem:

A group $G$ is said to be residually solvable if for every $g\in G \backslash \left\{e\right\}$ there is a normal subgroup $N$ of $G$ such that $g\notin N$ and $G/N$ is solvable.

Show that a finite residually solvable group is solvable.

Thanks in advance for your help.
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October 25th, 2017, 09:49 AM   #2
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By hypothesis, the obvious homomorphism of G into the direct product of all solvable factor groups of G is a monomorphism. Hence G is solvable.
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