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        Questions tagged [gr.group-theory]

        Questions about the branch of abstract algebra that deals with groups.

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        78 views

        Fourier transforms on finite abelian groups [on hold]

        Is the Fourier transform of a product of finite abelian groups equivalent to the product of the transforms, i.e. $$ \hat{f}\big(\prod_i G_i\big) = \prod_i \hat{f}(G_i),$$ where the $G_i$ are finite ...
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        85 views

        On covers of groups by cosets

        Suppose that ${\cal A}=\{a_sG_s\}_{s=1}^k$ is a cover of a group $G$ by (finitely many) left cosets with $a_tG_t$ irredundant (where $1\le t\le k$). Then the index $[G:G_t]$ is known to be finite. In ...
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        79 views

        Conjugacy in metaplectic groups

        Let $F$ be a non-Archimedean local field (characteristic 0) and $G=GL(2,F)$. Let $\tilde{G}$ be "the" metaplectic double cover of $G$ (defined using an explicit cocycle as in Gelbart's book (Weil's ...
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        2answers
        281 views

        Down to earth, intuition behind a Anabelian group [on hold]

        An anabelian group is a group that is “far from being an abelian group” in a precise sense: It is a non-trivial group for which every finite index subgroup has trivial center. I would like to know ...
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        53 views

        Unit-product sets in finite decomposable sets in groups

        A non-empty subset $D$ of a group is called decomposable if each element $x\in D$ can be written as the product $x=yz$ for some $y,z\in D$. Problem. Let $D$ be a finite decomposable subset of a ...
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        1answer
        84 views

        A converse of Cartan's automatic continuity theorem

        Let $G$ be a compact real Lie group. We say that $G$ has property $(*)$ if every abstract automorphism of $G$ is continuous. A theorem of Cartan says that if $G$ has perfect Lie algebra, it has ...
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        105 views

        Milnor's conjecture on Lie group (co)homology and forgetful functor of extensions

        Let $G$ and $H$ be compact Lie groups, Consider $Ext_{Lie}(G,H)$ the set of isomorphism of extensions of Lie groups: $$ 1\rightarrow G\rightarrow M\rightarrow H\rightarrow 1 $$ There exists a ...
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        1answer
        100 views

        Diameter for permutations of bounded support

        Let $S\subset \textrm{Sym}(n)$ be a set of permutations each of which is of bounded support, that is, each $\sigma\in S$ moves $O(1)$ elements of $\{1,2,\dotsc,n\}$. Let $\Gamma$ be the graph whose ...
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        68 views

        Any f.p. faithful simple module over a primitive group ring?

        Recall that a ring $R$ is primitive if it has a faithful simple left module. Let $G$ be a countable discrete group and $R=\mathbb{k}G$, where $\mathbb{k}$ denotes some field or $\mathbb{Z}$. There ...
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        1answer
        195 views

        Extensions of compact Lie groups

        Let $G$, $H$ be two compact Lie groups (possibly disconnected). Two short exact sequences of compact Lie groups $$ 0\rightarrow G\rightarrow M_1 \rightarrow H\rightarrow 0, $$ $$ 0\rightarrow G\...
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        1answer
        53 views

        Example of primitive permutation group with a regular suborbit and a non-faithful suborbit

        I would like some examples of groups $G$ satisfying all of the following criteria: $G<Sym(n)$, the symmetric group on $n$ letters, and $G$ is primitive. $G$ has a regular suborbit, i.e. if $M$ is ...
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        1answer
        99 views

        Characters of orthogonal groups as symmetric functions

        This question was asked on MSE some time ago, here, but got no attention. The Schur functions are characters of irreps of the unitary group, $s_\lambda(U)=Tr(R_\lambda(U))$. They are symmetric ...
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        1answer
        126 views

        Even Counterexample to Statement About the Non Existence of Certain Groups with Two Irreducible Monomial Character Degrees

        Let $\textrm{cd}(G)=\lbrace \chi(1)\,|\, \chi\in\textrm{Irr}(G)\rbrace$ denote the set of character degrees of a finite group $G$. Similarly, denote by $\textrm{mcd}(G)$ the set of monomial character ...
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        129 views

        Is every group a subgroup of a centerless group with the same prime order elements?

        Suppose $G$ is a group. Is $G$ a subgroup of some group $H$ such that: $H$ is centerless; If $h \in H$ is an element of prime order $p$, then there is also some $g \in G$ of order $p$. In other ...
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        1answer
        205 views

        Is residual finiteness a quasi isometry invariant for f.g. groups?

        A "residually finite group" is group for which the intersection of all finite index subgroups is trivial. Suppose $G$ and $G'$ are two quasi-isometric finitely generated groups. Does the residual ...

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