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4
votes
1answer
39 views
solvable subalgebra
I want to show that a set $B\subset L$ is a maximal solvable subalgebra.
With $L = \mathscr{o}(8,F)$, $F$ and algebraically closed field, and $\operatorname{char}(F)=0$ and
$$B= ...
3
votes
1answer
38 views
Is meta-$p$-nilpotent for all $p$ the same as meta-nilpotent?
I've been working on understanding solvable groups one prime at a time, and it has been very helpful. Perhaps this question is enjoyable to more than just me:
Definition: A finite group is said to be ...
3
votes
2answers
48 views
Index of maximal subgroups of $p$-solvable groups
If $G$ is finite $p$-solvable group (every chief factor has order either a power $p$ or relatively prime to $p$) must every maximal subgroup have index either a power of $p$ or relatively prime to ...
1
vote
1answer
40 views
Does the class of soluble groups whose $p$-length is $\leq 1$ for all $p$ form a saturated Fitting formation?
Let $\mathcal{F}$ be the class of all soluble groups $G$ such that the $p$-length $l_{p}(G) \leq 1$ for all primes $p$. How do I show $\mathcal{F}$ is a saturated Fitting formation?
6
votes
0answers
29 views
Cover and avoidance properties of chief factors: can we avoid exactly the ones we want?
If $G$ is a finite solvable group with chief series $1 = G_0 \leq \ldots \leq G_n = G$ (so each $G_i \unlhd G$ and if $G_{i-1} < H < G_i$ then $H$ is not normal in $G$) and $I \subseteq \{ ...
1
vote
2answers
68 views
If all Sylow subgroups are normal then the group is solvable
Show: all $p$-Sylowgroups are normal subgroups $\implies$ group $G$ is solvable.
I know that all subgroups of the different $p$-Sylowgroups are solvable, but do not know if this helps.
Other ...
3
votes
3answers
54 views
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 ...
2
votes
1answer
26 views
Number of Sylow bases of a certain group of order 60
We have $G=\langle a,b \rangle$ where $a=(1 2 3)(4 5 6 7 8)$, $b=(2 3)(5 6 8 7)$. $G$ is soluble of order 60 so has a Hall $\pi$ subgroup for all possible $\pi\subset\{2,3,5\}$. $\langle a\rangle$ is ...
4
votes
1answer
58 views
sylow basis of finite solvable groups
Let $G$ be a finite solvable non-$p$-group and $A$ be a maximal subgroup of $G$.
Therefore $A$ is of primary index $p^{n} $, that is $|G : A|=p^{n}$ where ...
0
votes
0answers
51 views
Solvable groups of order less than 168
I am asking myself the following question:
What are the solvable groups of order less than 168 ?
First of all, the smallest non-abelian simple group is $A_5$ (order $60$). Let $H$ be a group of ...
0
votes
1answer
43 views
Projective linear group - solvable
Let $q\geq 5$ and let PGL(2,q) be the projective general linear group.
Question
Do there exists a $q$ such that PGL(2,q) is solvable?
4
votes
1answer
131 views
Group of order $8p$ is solvable, for any prime $p$
Consider the following question:
Show that a group $G$ of order $8p$ is solvable, for any prime $p$.
I am kind of stuck, but here are my first attempts:
I chose the series of subroups ...
0
votes
2answers
58 views
Finite Group soluble $\iff G^{(k)}=\{e\}$ for some $k$
I am trying to understand the proof of the following proposition:
A finite group $G$ is soluble $\iff G^{(k)}=\{e\}$ for some $k$, where $G^{(0)}=G$, $G^{(i+1)}=[G^i G^i]$ is the derived series.
...
8
votes
2answers
182 views
Free groups in some classes?
I understand that the only free groups that are abelian are 1 and Z, hence a difference between 'free abelian groups' and 'abelian free groups'.
Can someone please tell me what are the solvable free ...
2
votes
1answer
43 views
Centralizer of the radical is a subgroup of the radical?
Let $G$ be a finite soluble group. We denote by $\mathfrak{N}$ the class of the finite nilpotent groups. The nilpotent radical, $G_\mathfrak{N}$, is the maximal subnormal nilpotent subgroup of $G$. I ...
13
votes
2answers
216 views
What can be said about two groups with isomorphic derived factors?
The third isomorphism theorem states that we can relate an isomorphic relation between two normal subgroups of a group $G$. My question is can we infer anything about the two groups structures itself ...
22
votes
1answer
255 views
Is the Galois group associated to a random polynomial solvable with probability 0?
Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$.
Let $r_1,\ldots,r_n$ be the roots of $P$ and consider ...
12
votes
4answers
260 views
If a finite group $G$ is solvable, is $[G,G]$ nilpotent?
If $G$ is a connected solvable Lie-group, then $[G,G]$ is nilpotent. The corresponding statement for Lie algebras follows from Lie's theorem, and it then follows from connected Lie groups by ...
2
votes
1answer
57 views
Is there a simple construction of a finite solvable group with a given derived length?
Given some integer $n$, is there an easy way to construct a finite solvable group of derived length $n$?
It would seem that given a solvable group of length $n-1$, one should be able to form the ...
10
votes
1answer
175 views
When is $G \ast H$ solvable?
In a proof that the lamplighter group $\mathbb{Z}_2 \wr \mathbb{Z}$ is not finitely presented, I showed that $\mathbb{Z}_2 \ast \mathbb{Z}$ is not solvable. More precisely, one can prove that the ...
