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

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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 ...
1
vote
0answers
27 views

Non-solvable, closed subgroups of $\mathrm{PSL}(2,\mathbb{R})$

It is mentioned here that non-solvable closed subgroups of $\mathrm{PSL}(2,\mathbb{R})$ are either the entire space or discrete. My question is this: Is there any easy proof of this, or do any of you ...
5
votes
4answers
277 views

Understanding non-solvable algebraic numbers

Background We know from Galois theory that the zeros of a polynomial with rational coefficients whose Galois group is solvable can be expressed in a formula that involves rational powers of the ...
5
votes
2answers
222 views

Why is a group of order 135 nilpotent?

Why does order 135 imply nilpotent?
0
votes
0answers
59 views

Express the exact root using radicals.

In Galois Theory, when a polynomial is not solvable by radicals, does that imply that the real solutions cannot be expressed exactly using the operators $+,-,\times,/$ and/or radicals?
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
82 views

Prove that if a group is nilpotent , Than it's quotient in its frattini subgroup is abelian.

I know that : 1) Nilpotent group is solvable. 2) Subgroup of a solvable group is solvable. 3) Solvable and simple group is abelian. Now I should use these facts to prove it.
2
votes
1answer
90 views

Show that the minimal polynomial of every element in $K=\mathbb{Q}(\zeta)$ is solvable by radicals, where $\zeta$ is a primitive 9th root of unity.

I have found the minimal polynomial if $\zeta$ over $\mathbb{Q}$ is $x^{6}+x^{3}+1$. $\mathbb{Q}(\zeta)\colon\mathbb{Q}$ is a normal and separable extension so ...
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.
0
votes
0answers
72 views

Every group of order 1000 is solvable

Here is how far I am: $1000 = 5^{3}\cdot 2^{3}.$ Number of $5$-Sylows is $1$ mod $5$ and has to divide $8,$ so its $1$ making it normal. Lets call it $F.$ Then $G/F$ has order $2^{3}$ and is thus ...
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
41 views

No group has commutator isomorphic to S_4

I'm wondering about an alternative solution to the following question from Dummit and Foote: Show that no group has commutator subgroup isomorphic to $S_4$. The hint says to use the previous ...
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.
0
votes
1answer
26 views

Smallest group with a derived series of length 2, 3 and 4

What are the smallest group with a derived series of length 2, 3 and 4?. I know that for n=2 the answer is S3 because that's the smallest metabelian non-abelian group. Could you help me out? Thanks
0
votes
0answers
16 views

Solvable finite group [duplicate]

If $G$ is a solvable group. Let the poset of subgroups of $G$ satisfy both the ascending and descending chain conditions. Then $G$ must be finite. Could someone please give me a hint? Thank you.
0
votes
0answers
43 views

Solvable group with ACC and DCC must be finite

Could some one please send me along a right path? I am unsure how to fit the "solvability" condition in this: If a solvable group's poset of subgroups satisfies both the ascending and descending ...
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 ...
0
votes
1answer
32 views

Sylow $q-$radical subgroup of a solvable group

Let ‎‎$G‎‎$ ‎be a‎ ‎finite ‎solvable ‎group ‎of ‎order ‎‎$p^2q^2‎‎$‎, where ‎$p>q‎$ ‎and ‎$‎q\nmid‎‎ ‎p-1‎$‎‎. ‎Let‎ ‎‎$G‎‎$ ‎has ‎the ‎following ‎presentation‎:‎ $‎‎‎\langle a‎ , ‎b‎ ,‎c \vert ...
2
votes
1answer
164 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 ...
1
vote
1answer
104 views

Show that$ f(x)=x^5-3$ is solvable by radicals over $\mathbb{Q}$.

I was reading about solvability of quintics by radicals, but unfortunately there were no many examples and I am afraid that I do not understand the whole concept. How to show $x^5-3$ is solvable by ...
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 ...
7
votes
2answers
206 views

Solvability by radicals of an equation of prime degree

For which prime $p$ the equation $x^p-p^px+p=0$ is solvable by radicals? I don't know how to solve this for primes $p\neq 2$, so any help is welcome.
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 ...
4
votes
1answer
55 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
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 ...