# Questions tagged [gr.group-theory]

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

**-1**

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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 ...

**2**

votes

**0**answers

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 ...

**3**

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**0**answers

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 ...

**3**

votes

**2**answers

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 ...

**1**

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**0**answers

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 ...

**3**

votes

**1**answer

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 ...

**2**

votes

**0**answers

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 ...

**4**

votes

**1**answer

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 ...

**5**

votes

**0**answers

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 ...

**3**

votes

**1**answer

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\...

**4**

votes

**1**answer

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 ...

**7**

votes

**1**answer

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 ...

**1**

vote

**1**answer

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 ...

**0**

votes

**0**answers

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 ...

**8**

votes

**1**answer

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 ...