6
votes
1answer
76 views

If $G$ is solvable and $G/[G,G]$ is cyclic, can $G\times G$ be generated by 2 elements?

I doubt this is true, but I haven't found any small counterexamples (there are no counterexamples with $|G| < 1536, |G| \neq 768$): Suppose $G$ is finite, solvable, 2-generated, and $G/[G,G]$ ...
0
votes
1answer
39 views

A question on solvable groups.

Let's suppose $S$ is a group and can be written as $S= G_1\times G_2\times\cdots\times G_K$; here each $G_i$ is a group of prime power order. And $o(S)= n.$ Is this $S$ solvable group? If yes how will ...
4
votes
1answer
45 views

Groups of order $p^aq^br^c$ containing elements of order $pq$, $qr$, and $pr$, but not $pqr$

Let $G$ be a solvable group of order $p^aq^br^c$ (for distinct prime $p,q,r$) containing elements of orders $pq$, $qr$, and $pr$, but no element of order $pqr$. Furthermore, assume that $G$ is ...
0
votes
0answers
36 views

Alt(6) is a subgroup of Alt(7)

How can it be shown that $Alt(6)$ is a subgroup of $Alt(7)$? Can we also prove the fact that $Alt(7)$ is not solvable?
1
vote
1answer
33 views

Proof for: semidirect product of solvable groups is solvable

Do you know the proof for the following statement or where I can find it? Semidirect product of solvable groups is solvable? I thought that this property is so ...
6
votes
1answer
117 views

Finite groups with all maximal subgroups of prime power index are solvable?

It is well known that in finite solvable groups, any maximal subgroup has prime power index. Question: If $G$ is a finite group in which any maximal subgroup has prime power index, is $G$ ...
2
votes
1answer
30 views

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 ...
5
votes
2answers
223 views

Why is a group of order 135 nilpotent?

Why does order 135 imply nilpotent?
2
votes
0answers
110 views

Maximal subgroups of direct product of solvable groups

Let $G_1$ and $G_2$ be finite solvable groups and $M$ be a maximal subgroup of $G=G_1\times G_2$ show that one of the following holds: $$ G_1\times\{e\}\leq M,\ \{e\}\times G_2\leq M,\ M \unlhd G.$$ ...
0
votes
2answers
50 views

If the product $x_1x_2…x_n=1$, prove that each $x_i=1$.

Let $G$ be a finite solvable group and let $x_1,x_2,...,x_n$ be elements of $G$ of pairwise relatively prime orders. If the product $x_1x_2...x_n=1$, prove that each $x_i=1$. I have no idea. Tell me ...
1
vote
1answer
40 views

Standard reference for Kaluzhnin's theorem

Kaluzhnin's theorem says that if $G$ is a group and $H \leq \operatorname{Aut}G$ acts trivially on each step of the normal series $1 = G_0 \leq G_1 \leq \ldots \leq G_n = G$, then $H$ has nilpotency ...
2
votes
1answer
77 views

Solvable and nilpotent groups, normal series and intuition

I'm reading Hungerford's algebra and I'm on Nilpotent and solvable groups chapter. Hungerford starts with: Consider the following conditions on a finite group G: i) G is the direct product ...
2
votes
2answers
66 views

Finite extensions of nilpotent groups

Let $G$ be a torsion free, nilpotent group and let $H$ be a normal subgroup of $G$ such that $G/H$ is a finite cyclic factor group. It is true that if $H$ is nilpotent of class $c$ then also $G$ ...
-1
votes
2answers
73 views

Prove that if a finite solvable group is simple, it is a cyclic group of prime order. [closed]

Prove that if a finite solvable group is simple, it is a cyclic group of prime order. Help me some hints.
3
votes
3answers
103 views

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.
3
votes
1answer
95 views

Solvable by radicals or not?

Let $f=2x^5-5x^4+5\in \Bbb Q[x]$. Then, how to prove that it's not solvable by radicals? Since $f$ is solvable by radicals iff $Gal(f)$ is solvable and $Gal(f)\subset S_5$ (up to isomorphism), I ...
2
votes
1answer
95 views

Infinite group with no maximal normal solvable subgroup

If $G$ is a finite group, then it has a maximal solvable normal subgroup. I think that this fails for infinite groups, but I don't know an example. Can you name one?
3
votes
1answer
62 views

Example of $A \le G$ solvable, $B \lhd G$ solvable, but $AB$ is not solvable

We'll denote $A \lhd G$ for $A$ a normal subgroup of $G$; and $A \leq G$ to mean $A$ is a subgroup of $G$. I don't know if it's a tricky question. But it seems strange to me. The question asks for an ...
1
vote
1answer
259 views

Every minimal normal subgroup of a finite solvable group is elementary abelian

Show that every minimal normal subgroup of a finite solvable group is an elementary abelian $p$-group for some prime $p$. I'm stuck on this one, any idea is appreciated.
3
votes
1answer
116 views

How to prove that a group of order $72=2^3\cdot 3^2$ is solvable?

Let $G$ be a group of order $$72=2^3\cdot 3^2$$ Without using Burnside's Theorem, how to show that $G$ is solvable? Atempt: If we can show that $G$ has at least one non-trivial normal subgroup $N$, ...
3
votes
1answer
66 views

Index of maximal proper subgroup of a solvable group

This is problem 2.7.15 from Hungerford's Algebra: If $H$ is a maximal proper subgroup of a finite solvable group $G$, then $[G:H]$ is a prime power. If $G$ is abelian, then it's easy to show ...
2
votes
1answer
165 views

Solvable group of certain order

‎We ‎know ‎that ‎in a non-abelian ‎group ‎of ‎order ‎‎$p^2q‎$ ($‎‎p$ ‎and ‎$q‎‎$‎ ‎are distinct primes)‎‎, ‎ if ‎$p>q‎‎$ and it's Sylow $p$-subgroup is elementary abelian $p$-group‎‎, ‎then ‎one ...
1
vote
0answers
101 views

Is the prime graph of a solvable group also the prime graph of some nonsolvable group?

Let $G$ be a solvable group. Does a nonsolvable group exist whose prime graph is isomorphic to the prime graph of $G$?
0
votes
1answer
78 views

$G$ be a non-nilpotent and supersoluble and 2-maximal subgroup of G permutes with all 3-maximal subgroup of G

Let $G$ be a non-nilpotent and supersoluble finite group. $G=Q\ltimes P$, Where $P$ is a group of order $p^{2}$($p$ is prime), all maximal subgroups of $P$ are normal in $G$, $Q=\langle a\rangle$ is ...
1
vote
0answers
75 views

$G$ be a non-nilpotent and supersoluble finite group.

Let $G$ be a non-nilpotent and supersoluble finite group. $G=Q\ltimes P$ and $|Q: C_{Q}(P)|=q$ is prime, Where $P$ is a group of prime order $p\neq q$. $Q$ is noncyclic group of order ...
4
votes
1answer
107 views

If $N$ and $G/N$ are virtually solvable, then $G$ is virtually solvable?

("Virtually solvable" means there's a solvable subgroup of finite index.) I know this statement holds if "virtually solvable" is replaced by "solvable", but I want to knock it down to the ...
1
vote
0answers
49 views

$G$ be a $p$-supersoluble group. and $O_{p'}(G ) = 1$ then $G$ is supersoluble

Let $G$ be a $p$-supersoluble group. If $O_{p'}(G ) = 1$ then $G$ is supersoluble.
2
votes
3answers
153 views

Is this group solvable?

Let $G=H\times K$ (a direct product of $H , K$) where $H$ is an abelian 2-group and $K$ is a non-abelian simple group. Is $G$ solvable? why? These three answers are true, and I thought so. But in a ...
2
votes
0answers
104 views

Class of $p$-supersolvable group is saturated formation.

I want to show that the class of $p$-supersolvable groups is a saturated formation. I only have the definition of $p$-supersolvable. What do I do? A group is $p$-supersolvable iff every chief factor ...
1
vote
0answers
83 views

Supersolvable group and nilpotent maximal subgroup

$G$ is a supersolvable group, $|G|=pq^{b}$ ($p$ and $q$ are different prime numbers). $Q$ is a $q$-Sylow subgroup of $G$, $Q=\langle x\rangle$ and $\operatorname{Cor}_{G}(Q)=\langle x^{q}\rangle$. If ...
3
votes
1answer
54 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
194 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 ...
3
votes
1answer
73 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?
7
votes
0answers
53 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
160 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
60 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
44 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
100 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
153 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
59 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?
5
votes
1answer
283 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
66 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. ...
13
votes
2answers
247 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 ...
2
votes
1answer
56 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 ...
11
votes
1answer
281 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 ...
12
votes
4answers
408 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 ...
8
votes
2answers
209 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 ...