Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
21 questions from the last 30 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 ...
- 1,219
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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' ...
- 2,603
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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 ...
- 638
6
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1
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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
3
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1
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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 ...
- 1,303
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A profinite group that is not a completion
Is there a known example of a profinite group $G$ such that $G$ is not the profinite completion of any residually finite group?
By Nikolov and Segal's work, an example must be not finitely generated.
- 151
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Closest standard framework for a cyclically ordered 8-set with a fixed-point-free involution?
I am studying the following finite structure.
Let
$$
R=\{2,3,4,5,6,7,8,9\}, \qquad
L=(2\,3\,4\,6\,5\,8\,7\,9), \qquad
\delta=L^4=(2\,5)(3\,8)(4\,7)(6\,9).
$$
So $R$ is equipped with a cyclic order (...
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2
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Is a "random" 2-generator 2-relation perfect group nontrivial?
Last week some younger grad students were asking for help with practice problems for their
topology final. One of the questions was accidentally much harder than the professor intended, in a way that ...
- 1,023
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1
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Is there a mapping from a free group to a surface group that preserves trace equivalence?
A couple of days ago I asked this question. The answer was very helpful and it has encouraged me to rework the question.
We say two elements $u$ and $v$ in a group ($G$) are "trace equivalent&...
- 675
3
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216
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Structure of residually soluble groups with nilpotent quotients
(This question is copied from Stack Exchange. It seems quite specific, so I hope an expert here might be able to provide a relevant technique.)
Let $G$ be a finitely generated, residually finite and ...
- 1,219
9
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1
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If two elements are trace equivalent in a surface group are they trace equivalent in a free group?
We say two elements $u$ and $v$ in a group ($G$) are "trace equivalent", $u \equiv_{\operatorname{tr}} v$, if for every complex representation, $\alpha : G\rightarrow \operatorname{SL}(2,\...
- 675
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Decomposing finite groups into unions of transversal normal subgroups
Problem. Assume that a finite group $G$ is the union $G=\bigcup_{i=1}^nH_i$ of $n\ge 2$ nontrivial normal subgroups $H_i$ such that $H_i\cap H_j=\{e\}$ for all distinct indices $i,j\le n$. Is $G$ ...
- 46k
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Isomorphism of composition factors of representatives of the Aschbacher classes
Let $G_1$ and $G_2$ be two classical groups of Lie type that are isomorphic as abstract groups but arise from different classical constructions (for instance, via different natural modules or forms).
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Spectrally undetermined finite orthogonal groups
For a representation $\rho$ of a group $G$, we denote by $\operatorname{tr}(\rho)$, the multiset $\{\{\operatorname{tr}(\rho(g)):g\in G\}\}$. Similarly, we denote by $\operatorname{sp}(\rho)$ the ...
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