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October 21st, 2017, 08:42 AM  #1 
Senior Member Joined: Jan 2015 From: usa Posts: 104 Thanks: 1  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. 
October 25th, 2017, 09:49 AM  #2 
Member Joined: Jan 2016 From: Athens, OH Posts: 93 Thanks: 48 
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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