Questions tagged [solvable-groups]

Question 1

Let $G$ be a virtually polycyclic group with subgroups $H$ and $K$. Let $h(\,.)$ denote the Hirsch length. Does the inequality $$h(G) \geq h(H) + h(K) - h(H \cap K)$$ always hold? (I've found this ...

Question 2

It is known that for any finite simple group $G$ there exist two elements $a,b\in G$ such that $a$ and $b$ are conjugates in $G$ and $\langle a, b \rangle=G$. My question: Is it true for any finite ...

Question 3

There is a theorem by Wang and Chen that says: when the finite group $A$ acts via automorphisms on the finite group $G$ with $|A|$ and $|G|$ coprime, and $C_G(A)$ is either odd-order or nilpotent, ...

Question 4

In Finite Group Theory by Isaacs, problem 3C.8 essentially asks: show two nilpotent injectors in a finite solvable group $G$ are conjugate. Isaacs uses a non-standard definition of nilpotent injector, ...

Question 5

Let $G$ be a finite non-nilpotent, solvable group such that its commutator subgroup $G'$ is the unique minimal normal subgroup of $G$. Is it necessarily true that $G/G'$ is cyclic? I believe it is not ...

Question 6

Let $G$ be a finite group, and let $\Phi(G)$ denote its Frattini subgroup. Since $\Phi(G)$ is contained in every maximal subgroup of $G$, it is natural to ask about the relationship between the normal ...

Question 7

Let $F$ be a finite field and let $G$ be the subgroup of upper triangular matrices in $GL_n(F)$. I am working through an exercise to show that $G$ is a soluble group. The exercise has the following ...

Question 8

Let $f(x)$ be a rational function over $\mathbb{Q}$. Suppose that $(x_0,f(x_0))$ is a local extremum of the function, so in particular, $x_0$ is a root of $f'(x)$. Is it possible for $f(x_0)$ to be ...

Question 9

I was reading the proof of Proposition 12.31 of Roberto Frigerio in "Bounded cohomology of discrete groups": Proposition $12.31$. Let $ρ: \Gamma g → \operatorname{Isom}^+(\mathbb{H}^2)$ be ...

Question 10

A positive integer $n$ is said to be a solvability-forcing number if every group of order $n$ is solvable. In here it is stated that $n$ is a solvability-forcing number if and only if no divisor of $n$...

Question 11

This is Exercise 5.18(i) from Rotman's An Introduction to the Theory of Groups. I had two ideas for trying to solve this exercise. The first is to take solvable series $S = S_0 \geqslant...\geqslant ...

Question 12

Let $G$ be a finite group such that $G =\langle x,y\rangle$ where $x$ and $y$ are conjugates in $G$. Does this necessarily imply that $G$ is solvable? P.S. I can not give any concrete reason behind my ...

Question 13

Lang’s Algebra in p.308 is written as follows, for any embedding $\sigma$ of $E$ in $E^*$ over $k$, extension $\sigma E/k$ is also solvable by radical, it is pretty brief Hence, my proof of this two ...

Question 14

Let $T(n,R) < \mathrm{GL}(n,R)$ be a group consisting of upper (or lower) triangular $n \times n$ matrices over a ring $R$. It is clear that when $R$ is a field, $T(n,R)$ is soluble. (Hint: ...

Question 15

Given $p>2$ prime, is there any finite solvable group with $mp+1$ Sylow $p$-subgroups with $1 \leq m < p$ such that $mp+1$ is neither a prime power nor $(qt)^a$ for $q,t$ primes? I'm looking for ...

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Questions tagged [solvable-groups]

An inclusion-exclusion-like inequality for the Hirsch length

Generating a finite non-solvable group with an element and its conjugate

Wang-Chen theorem on solvability?

Solving Isaacs FGT problem 3C.8 using Carter subgroups

A Query on Commutator Subgroup

Can the normal subgroups of $G \over \Phi(G)$ be determined from those of $G$

Approach to showing that the group of invertible upper-triangular matrices over a finite field is soluble [duplicate]

Can a local extremum be solvable by radicals without the critical point being so?

Elementary subgroups of $\operatorname{Isom}^+(\mathbb{H}^2)$ are solvable.

Classification of the solvability-forcing numbers

If $S$ and $T$ are solvable subgroups of $G$ with $S\trianglelefteq G$, then $ST$ is solvable.

A group generated by an element and its conjugate must be solvable.

Brief proof of Embedding to algebraic closure perserve solvable by radical

Supersolvability of some triangular groups

Any finite solvable group with $mp+1$ $p$-Sylows, $1 \leq m<p$, $p>2$ such that $mp+1$ is neither $q^a$ nor $(qt)^a$ for some $q,t$ primes?

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