
Abstract Algebra Abstract Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
October 21st, 2017, 09:42 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, 10:49 AM  #2 
Member Joined: Jan 2016 From: Athens, OH Posts: 92 Thanks: 47 
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.


Tags 
group, residually, solvable 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Last nontrivial group in the derived series of a solvable group  fromage  Abstract Algebra  0  December 4th, 2016 03:39 AM 
solvable Group of matrices over Z_p  Sandra93  Abstract Algebra  7  November 4th, 2013 01:05 PM 
is it solvable ?  versmart  Algebra  8  September 15th, 2011 08:07 AM 
Minimal Normal subgroup of a solvable group  SonicYouth  Abstract Algebra  3  March 13th, 2011 01:31 AM 
is it solvable ?  versmart  Abstract Algebra  5  December 31st, 1969 04:00 PM 