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

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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= ...
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 ...
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 ...
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 \{ ...
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: $$ G_1\times\{e\}\leq M,\ \{e\}\times G_2\leq M,\ M \unlhd G.$$ ...
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
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 ...
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 $G$?
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
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.
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 ...
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?
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?
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 ...
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 ...
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 ...