All Questions
12 questions
0
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2
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363
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proof that if G is a solvable group then quotient of G is solvable
I am reading Stewart book on Galois theory, one of the theorem proves that if $G$ is solvable and $N$ is a normal subgroup of $G$, then the quotient of $G$, $G/N$ is solvable. I can't fully understand ...
- 147
4
votes
1
answer
149
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Solvable-by-finite groups
I am trying to prove this:
Let $G$ be a finite-by-solvable group, i.e. $G$ has a normal subgroup $N$ that is finite with $G/N$ solvable. Prove that $G$ is solvable-by-finite, i.e., $G$ has a solvable ...
- 69
1
vote
1
answer
84
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Solvable normal subgroup but the corresponding quotient group is not solvable [closed]
Let $G$ be a group and $N$ be a normal subgroup of $G$.
It is a well known fact that: $G$ is solvable iff $N$ and $G/N$ are solvable.
I wonder the following:
Could you give an example of a group $G$ ...
- 457
2
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0
answers
44
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Problem (12.4) in Isaacs Character theory of finite groups
I am trying to solve problem (12.4) in Isaacs' Character theory of finite groups.
The problem is to show that if $G$ is non-abelian solvable and has no non-abelian factor group of prime power order, ...
- 21
1
vote
0
answers
60
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What is the quotient group $G/R(G)$?
The maximal solvable normal subgroup of $G$ is called the radical subgroup of $G$, and it is denoted by $Rad(G)$ of $R(G)$.
Question: Is the quotient group $G/R(G)$ been classified (i.e., it always ...
- 564
0
votes
0
answers
101
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How to show that if $G/N$ and $N$ soluble then so is $G$ [duplicate]
I am trying to show that if $N$ is a normal subgroup of $G$ that is soluble and $G/N$ is soluble then so is $G$.
I am wondering whether it has something to do with the isomorphism theorem for groups. ...
- 1,880
0
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0
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116
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Showing Factor Groups are Simple; Subgroups of Solvable Groups are Solvable
I'm aware this post is related to many other posts on proving that subgroups of solvable groups are solvable. However, there's a certain claim that is necessary to show based on the definition of ...
- 1,259
0
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1
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348
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Understanding a proof of: If $N\unlhd G$ s.t. $N$ and $G/N$ are solvable, then $G$ is solvable.
Let $N$ be a normal subgroup of $G$ s.t. $N$ and $G/N$ are solvable, then $G$ is solvable.
Proof:
Because $N$, $G/N$ are solvable $\Rightarrow$ $N^{(s)}=\{e\}$, $(G/N)^{(t)}=\{e\}$ for $s,t\in \mathbb{...
- 1,604
0
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0
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146
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Quotient $G/N$ of solvable group $G$ is solvable. Is my (trivial) proof correct?
I'd like some feedback from experts in group theory.
Let $N \trianglelefteq G$, $G$ solvable. Then $G/N$ is solvable.
My proof: Take canonical homomorphism $G \rightarrow G/N$ ($x \rightarrow xN$) and ...
- 329
3
votes
1
answer
383
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show that if $G/K, G/N$ are solvable then $G/(K \cap N)$ is also solvable
Let $G$ be a group and let $N,K \lhd G$ be her normal subgroups. It is given that $G/N$ and $G/K$ are solvable.
I need to prove that $G/(K \cap N)$ is also solvable.
I thought about proving somehow ...
- 2,031
0
votes
1
answer
110
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Basic theorem for solvable groups not true for nilpotent groups - counterexample.
it's my first question on MathStackExchange so please be tolerant.
Let H be a normal subgroup of group G. If H and G/H are both solvable, then G is solvable. But H nilpotent and G/H nilpotent doesn't ...
- 904
7
votes
1
answer
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Proof of $G$ is solvable implies $G/N$ is solvable.
I want to show that if $N$ is normal in $G$ then $G$ is solvable implies $G/N$ is solvable.
Now, $G$ is solvable implies there exists a chain
$\{e\}=G_0 \trianglelefteq G_1 \trianglelefteq G_2 \...
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