Tagged Questions

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

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Why is a finite group, with a maximal subgroup which is abelian, soluble?

I've come across an exercise saying the following: If $G$ is a finite group which contains a maximal subgroup $M$ which is abelian, show that $G$ is solvable and that $G^{(3)}$ (the third term in the ...
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A proof that BS(1,2) is not polycyclic

I am looking for examples of finitely generated solvable groups that are not polycyclic. In Wikipedia Baumslag-Solitar group $BS(1,2)$ is an example. But how to prove this fact?
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Existence a certain subgroup of a group

‎Let ‎‎$G$ ‎be a‎ ‎finite group ‎such ‎that ‎‎$G=P\rtimes Q‎‎‎‎$ ‎where ‎‎$P\in {\rm Syl}_p(G)‎‎$;‎ ‎‎$‎‎P\cong \Bbb{Z}_p\times \Bbb{Z}_p‎$ ‎‎‎‎ and ‎$Q\in {\rm Syl}_q(G)$; ‎$‎‎|Q|=q$ (‎$‎‎p, q$ ‎are ...
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Finding a particular chief series for a supersolvable group

Let $G$ be a finite supersolvable group. Since it is supersolvable it has (by definition) some normal series $1=N_0 \le N_1 \le \cdots \le N_n=G$ such that each factor $N_i/N_{i+1}$ is cyclic. Also ...
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A well-known theorem of O. Schmidt

Prove that if all the proper subgroups of a finite group $G$ are nilpotent, then $G$ is soluble. How to I prove it? Thanks in advance.
prove that $GL_2(\Bbb Z_3)$ is solvable
I need to prove that $GL_2(\Bbb Z_3)$ is solvable What I tried: I know that $GL_2(\Bbb Z_3)$ has $(3^2-1)(3^2-3) = 48 = 3 * 2^4$ elements. I know that $n_3 \in \{1,4,16\}$ and $n_2 \in \{1,3\}$ ...