# Questions tagged [solvable-groups]

For questions on solvable groups, their properties, and structure.

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### Does every normal subgroup of a finite solvable group appears in some subnormal series whose factor groups are all abelian? [duplicate]

Let $G$ a finite solvable group. Does every normal subgroup of $G$ appears in some subnormal series whose factor groups are all abelian? Any comment is welcome.
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### When does this Galois theorem "variation" holds?

Im reading about solvability by radicals in different books, more concretely in Fields and Galois Theory by Patrick Morandi and Introduction to abstract algebra by Benjamin Fine, etc. In the second ...
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### Clarification over subnormal series of a group

I am reading through a lecture note on group theory and it says the following: Let $H \leq G$ be a subgroup of group $G$. If $G_n \trianglelefteq \cdots \trianglelefteq G_1 = G$ is a subnormal tower ...
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### prove a finite nilpotent group has supersolvable series using upper center series only

Let $G$ be a finite nilpotent group. If G has a normal series $$\langle e\rangle=G_0\leq G_1\leq ... \leq G_n=G$$ such that $G_i\lhd G$ and $G_{i+1}/G_i$ is cyclic for all $i$, then G is called ...
63 views

### Does Galois theory provide a method for solving solvable polynomials?

A polynomial $f(x)$ of degree larger than $5$ may not be solvable. But if we take, for example, $$f(x)=x^8+x^6+x^4+2,$$ we can see by introducing a new variable $u=x^2$ (by observation) and it ...
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### A matrix group over commutative ring

Let $R$ be a commutative ring and denote by ${\rm GL}_n(R)$ the general linear group over $R$. Let $I_n$ denote the identity matrix of size $n$. Let $G$ be a subgroup of ${\rm GL}_n(R)$ such that ...
86 views

### 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 ...
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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 ...
### Let $G$ be a group such that $N\unlhd G$ and $H \le G$ are both solvable. Show that $NH$ is solvable. [duplicate]
Since, $N \unlhd G$ and $H \le G$, from the second isomorphism theorem we know that $N \unlhd NH$, therefore $NH$ is solvable if $N$ and $NH/N$ are solvable. From our assumptions, $N$ is solvable, ...