4
votes
1answer
59 views

Unique normal subgroups of every possible order

Can one characterize groups $G$ for which there is a unique normal subgroup of order $d$ for every divisor of the order of $G$? For example must they be solvable?
2
votes
1answer
52 views

Characterization of solvable groups in terms of subgroups of certain orders?

In this question, the OP mentions the following result: a finite group $G$ is solvable if and only if $$\text{for all $n$ dividing $|G|$ such that $\gcd(\frac{|G|}{n},n)=1$, $G$ has a subroup order ...
1
vote
2answers
43 views

nilpotent group implies solvable group

Can someone please do a simple proof of this: If a group, G, is nilpotent then it is solvable. I'm pretty bad at math and am just trying to figure this out. Thank you very much!
2
votes
1answer
62 views

On the Hall Theorem

I went to search info about it in books (as Isaac), this site, wikipedia. But no one says nothing about my (maybe silly) doubt. The theorem states that, given a finite and solvable group $G$, and an ...
6
votes
1answer
84 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 $G/[G,G]$ ...
4
votes
1answer
60 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 ...
6
votes
1answer
201 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
39 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 ...
2
votes
0answers
157 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
53 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
votes
2answers
102 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
109 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.
2
votes
1answer
48 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
111 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?
1
vote
1answer
354 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
30 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
1answer
35 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
176 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 ...
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
76 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
0answers
52 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
0answers
117 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
88 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
58 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
210 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
80 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
60 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 \{ ...
3
votes
3answers
63 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
48 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
108 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
1answer
67 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?
0
votes
2answers
71 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. ...
2
votes
1answer
59 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 ...
3
votes
1answer
82 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 ...
25
votes
1answer
345 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
440 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 ...