Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
6 questions from the last 7 days
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Let G be a finite group such that for every natural n the number of solutions to x^n=e is smaller/equal to n. Prove G is cyclic [migrated]
I received this question as homework for a graduate-level course. I don't want the full answer, just a hint on how to proceed with my current direction.
I reduced the problem to the following case: ...
- 11
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Structure of residually nilpotent groups with nilpotent quotients
Let $G$ be a finitely generated, residually finite, and residually nilpotent group. Suppose $G$ satisfies the following properties:
Every proper quotient of $G$ is virtually nilpotent with Hirsch ...
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The principal block of a Frobenius group
Let $G$ be a finite group. Fix a prime $p$. Let $P$ be a Sylow $p$-subgroup such that $P\cap P^x=1$ for all $x\not\in P$. (In other words, $P$ is a Frobenius complement.)
It follows from Frobenius' ...
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A query about factorizable simple groups
Let $G$ be a finite simple group and $M_1,M_2,M_3,M_4$ be four distinct non-conjugate maximal subgroups of $G$ such that $M_1M_2=G=M_3M_4$. I feel the following is true:
There exist $i\in \{1,2\}$ and ...
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6
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F.g. graphs of groups with f.g. edge groups have f.g. vertex groups
I am looking for a reference for the following result:
Let $\mathbb X$ be a graph of groups whose underlying graph $X$ is finite and whose edge groups are finitely generated. If the fundamental group $...
- 1,135
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What is this 'topological-cycle-space' weakening of the finiteness property $\mathrm{FP}_2$?
$\newcommand{\FP}{\mathrm{FP}}$Let $G$ be a finitely generated group and $X$ a locally finite, simplicial Cayley graph of $G$. Let $\mathscr C(X)$ denote the cycle space of $X$. In other words, this ...
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