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        Questions on group theory which concern finite groups.

        6
        votes
        0answers
        61 views

        Numbers where there is a unique group with integral character table

        Call a number $n$ special in case there is a unique group of order n whose character table consists of only integers. This should be equivalent to the condition that the number of conjugacy classes ...
        25
        votes
        1answer
        492 views

        Number of irreducible representations of a finite group over a field of characteristic 0

        Let $G$ be a finite group and $K$ a field with $\mathbb{Q} \subseteq K \subseteq \mathbb{C}$. For $K=\mathbb{C}$ the number of irreducible representations of $KG$ is equal to the number of conjugacy ...
        11
        votes
        3answers
        430 views

        Which partitions realise group algebras of finite groups?

        Fix an algebraically closed field $K$ (maybe of characteristic zero first for simplicity, like $\mathbb{C}$). Given a partition $p=[a_1,...,a_m]$ of an integer $n$. We can identify $p$ with the ...
        2
        votes
        0answers
        96 views

        How large can a symmetric generating set of a finite group be?

        Let $G$ be a finite group of order $n$ and let $\Delta$ be its generating set. I'll say that $\Delta$ generates $G$ symmetrically if for every permutation $\pi$ of $\Delta$ there exists $f:G\...
        3
        votes
        1answer
        117 views

        Littlewood Richardson Rule for general linear group over finite field

        I just finished reading Green's 1955 paper on characters of general linear groups and have also been reading Macdonald's Symmetric Functions and Hall Polynomials. I see that there is a recursive ...
        6
        votes
        1answer
        213 views

        Subsets of a group with special property

        Let $G$ be a finite group. We say a subset $A$ of $G$, $|A|=m$, is $(m,i)$-good, $m\geq 1$ and $0\leq i\leq m$, if there exist $g_A\in G$ such that we have $|gA\cap A|=m-i$. I need some groups such ...
        1
        vote
        0answers
        120 views

        Does there exist an unramified $PSL_2(\mathbb{F}_p)$ extension of a quadratic field $K$?

        Does there exist an unramified $PSL_2(\mathbb{F}_p)$ extension of a quadratic field $K$ for $p\geq 5$?
        4
        votes
        0answers
        153 views

        Is the following variant of Shafarevich's theorem known?

        Let $Q$ be a finite simple group which may be realized as the Galois group of some extension of $\mathbb{Q}$ (like for instance $PSL_2(\mathbb{F}_p)$ for $p\geq 5$, or the monster group) and let $G$ ...
        3
        votes
        1answer
        71 views

        Bounding the derived length of a solvable group given the degrees of the irreducible monomial characters

        Much is known about the derived length of a solvable group given the degrees and cardinality of the set of degrees of the irreducible characters. Martin Isaacs and Donald Passman pretty much started ...
        3
        votes
        1answer
        107 views

        Conditions for a solvable group to have a non-trivial center

        I am working on a problem in character theory where I try to bound the derived length of a solvable group using information about its characters. In my specific case, it will be extremely helpful for ...
        3
        votes
        0answers
        48 views

        Zero divisors with support size 3 in complex group algebras of residually finite groups

        Conjecture. There exists a function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that if $\beta$ is a non-zero element of the complex group algebra $\mathbb{C}[G]$ of a finite group $G$ such that $1\...
        6
        votes
        1answer
        170 views

        Zero divisors in complex group algebras of residually finite groups

        Conjecture. There exists a function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that if $\alpha$ and $\beta$ are non-zero elements of the complex group algebra $\mathbb{C}[G]$ of a finite group $G$ ...
        10
        votes
        1answer
        168 views

        Low dimensional representations of $SL_n(\mathbb{Z}/p^\ell \mathbb{Z})$

        When $\ell = 1$ I know that the smallest non-trivial irreducible complex representations of $SL_n(\mathbb{Z}/p\mathbb{Z})$ has dimension $\frac{p^n - 1}{p-1} - 1$ (with maybe some exceptions for ...
        4
        votes
        0answers
        98 views

        Examples for Bogomolov multiplier of finite group

        Take $G$ to be a finite group. The projective representations of $G$ are classified by the group cohomology $H^2(G,U(1))$. Here $G$ has a trivial action on $U(1)$. We focus on the restriction map $H^...
        1
        vote
        0answers
        78 views

        What is an upper limit of relative size of conjugacy class of the transitive finite group?

        What is $$ \limsup_{n\to\infty} \sup_{\deg(G)=n} \left( \frac{\max_{x\in G} \left|\operatorname{conj}(x)\right|}{\operatorname{ord}(G)}\right),$$ $G$ transitive permutation group? And what are the ...

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        山西福彩快乐十分钟
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