# Questions tagged [schur-multipliers]

The schur-multipliers tag has no usage guidance.

28
questions

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### Stem extensions and quotients of Schur covers

Suppose that $G$ is a finite group, and that $\Gamma$ is a central extension of $G$ by $A$, that is
$$ 1 \rightarrow A \rightarrow \Gamma \rightarrow G \rightarrow 1$$
with the image of $A$ contained ...

5
votes

1
answer

312
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### Surjection onto $H_{2}(\mathrm{PGL}(2,\mathbb{C}),\mathbb{Z})$

Let $G \leq \mathrm{PGL}(2,\mathbb{C})$ be the subgroup of upper-triangular matrices. I am interested in the natural morphism on the Schur multiplier (i.e. group homology as discrete groups)
$H_{2}(G,...

3
votes

1
answer

294
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### Schur multiplier of $\mathrm{SL}(2,\mathbb{Q})$

What is the Schur multiplier of $\mathrm{SL}(2,\mathbb{Q})$? The techniques used here give $K_2(\mathbb{Q})$ as a lower bound, but it’s probably bigger than that, especially since the universal cover ...

8
votes

1
answer

436
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### Is there a algorithm to compute the Schur multiplier of a finite group from a group presentation

Suppose we have a finite group $G$ whose presentation or Cayley table is given. Is there an algorithm (at least theoretically - without considering computational complexity) to compute the Cayley ...

9
votes

1
answer

530
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### What is the Schur multiplier of the Mathieu group $M_{10}$

It is well known that the automorphism group of the alternating group $A_6$ is $P\Gamma L_2(9)$. There are three different index $2$ subgroups of $P\Gamma L_2(9)$, namely the symmetric group $S_6$, ...

7
votes

2
answers

559
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### Representation of central extension

Let $G$ be a finite abelian group of rank $n$ and $H\rightarrow G$ a central extension with cyclic finite kernel.
Is it true that we can find a faithful representation $H\rightarrow {\rm GL}_{k(n)}(\...

0
votes

0
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79
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### What do Sylow 2-subgroups look like for Schur covering groups of finite simple groups?

What do Sylow 2-subgroups look like for Schur covering groups of finite simple groups?
Are there any references in which we can find the stucture of Sylow 2-subgroups of Schur covering groups of ...

1
vote

0
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61
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### Reference request for finite simple exceptional group of lie type $E_7(q)$ and its Schur covering group $2.E_7(q)$?

Does anyone have the paper named 'Génerateurs, relations et revêtements de groupes algébriques' written by Robert Steinberg in 1962, or any other reference for simple groups of Lie type $E_7(q)$ and ...

3
votes

0
answers

66
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### Is there a method to find the order of a lift of an element of order 2 to the Schur cover?

Let $G$ be a finite non-abelian simple group, $M.G$ the Schur covering group of $G$. Is there a method to find the order of a preimage of an element of order 2 in the natural homomorphism $\pi: M.G\...

2
votes

2
answers

249
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### In which books we can find structure information for finite simple groups and their Schur covering groups?

In which books we can find representations or character tables, Sylow 2-subgroups and conjugacy classes for finite simple groups and their Schur covering groups and properties for Schur multiplier of ...

4
votes

2
answers

427
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### Is $1\neq a\in Z(2.E_7(q))\cong Z_2$ a square element in $2.E_7(q)$?

When $q$ is a power of some odd prime, is $1\neq a\in Z(2.E_7(q))\cong Z_2$ a square element in $2.E_7(q)$?
A Lie algebra is a vector space $L$ over a field $K$ on which a product operation $[xy]$ is ...

9
votes

1
answer

306
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### Reference for Schur multiplier identity

Let $G$ be a finite group and $H$ a normal subgroup of $G$. I recently stumbled upon the following identity for the Schur multiplier of $G/H$:
$$\operatorname{H}_2(G/H,\mathbb{Z}) \cong \frac{\...

9
votes

1
answer

262
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### 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^...

5
votes

1
answer

305
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### Schur covers of affine 2-transitive groups

I am interested in Schur covers of minimal 2-transitive groups. A theorem of Burnside gives that every finite 2-transitive group is either almost simple or affine. In the time since, these groups have ...

2
votes

1
answer

140
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### Convergence of sequence of images of Schur multipliers

Let $\eta$ be a continuous bounded function on $(0, \infty)^{2}$ so that $\eta(0,0)=1$. Let $A$ be a bounded operator on $\ell^{2}(\mathbb{Z}_{\geq 0})=\ell^{2}$ (by bounded operator I will always ...

2
votes

1
answer

139
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### Kernel of a double cover of group as stem extension

A stem extension of a group $X$ is a group $G$ with a subgroup $N$ contained in $G' \cap Z(G)$ such that $G/N$ is isomorphic to $X$, so that we have a short exact sequence $1 \to N \to G \to X \to 1$.
...

4
votes

0
answers

131
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### When is the restriction map $res:H^2(G,U(1))\to H^2(Z_p\times Z_p,U(1))$ not the zero map?

Consider $G$ to be a finite group with non-trivial Schur Multipler $H^2(G,U(1))$, where $G$ acts trivially on the circle group $U(1)$.
By Example of a Schur-nontrivial group with no abelian subgroup ...

1
vote

1
answer

195
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### Example of a Schur-nontrivial group with no abelian subgroup of the form $H\times H$?

A group $G$ is Schur-nontrivial if the Schur multipler $H^2(G,U(1))$ is not the trivial group.
I am trying to find an example of a Schur-nontrivial group which does not contain a subgroup of the form ...

1
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0
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247
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### When can a 2-cocycle on a subgroup can be extended?

This question is based on a question when is the restriction $H^2(G,\mathbb{C}^*)\to H^2(K,\mathbb{C}^*)$ surjective?
I am asking this as a new question as I already asked that user but got no ...

2
votes

1
answer

82
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### How to claculate the $T$-stable subgroup of second cohomology group

Let $G=\langle x,y,z,w \mid [y,x]=w^p=z, x^{p^2}=y^p=z^p=1 \rangle$ be a group, where $[u,v]=u^{-1}v^{-1}uv$, $p$ is a prime and the commutator which do not appear is 1.
Let $N=\langle y,w \rangle \...

4
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0
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### Is the Tensor/Exterior square $G\otimes G$ or $G\wedge G$ of infinite p-group also a p-group?

Let $G$ be an infinite countable p-group. Is it true that $G\otimes G$ or $G\wedge G$ are also p-groups? (where G acts on itself by conjugation). For simplicity, you can consider that $G=[G,G]$, and ...

8
votes

1
answer

493
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### What is the Schur multiplier of the affine linear group AGL(n,q)?

What is the Schur multiplier of the $n$-dimensional affine linear group $\mathrm{AGL}(n,q)$ over the Galois field with $q$ elements?
I am particularly interested in the simple case $n=1$. Computation ...

1
vote

1
answer

373
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### Two questions on the Schur multiplier of groups of order $p^4$

I tried to find a reference for the computation of the Schur multiplier of groups of order $p^4$.
The case in which $p=2$ is well known, see e.g. Table 1 at http://pages.bangor.ac.uk/~mas010/pdffiles/...

1
vote

0
answers

105
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### Homology and Exterior Square

Let $G$ be a finite group and $G \wedge G$ denote the exterior square of $G$. It is well known that the second integral homology $H_2(G,\mathbb{Z})$ is the kernel of homomorphism $x \wedge y \mapsto [...

13
votes

2
answers

761
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### How hard (P, NP, NP-hard) is it to compute Schur norms of matrices (as multipliers)?

Given a matrix $A\in M_n(\mathbb{C})$, I will denote by $||A||_\infty$ the operator norm of $A$, as seen acting on the Hilbert space $\mathbb{C}^n$. This makes $M_n(\mathbb{C})$ into a Banach space (...

0
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0
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148
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### A question about multiplier algebra of $C_0(G)\otimes C_b(G)$ for a locally compact group $G$

Let $G$ be locally compact group. How we can show that
$$
M(C_0(G)\otimes C_b(G))=C_b(G,C_b(G)).
$$
($M(C_0(G)\otimes C_b(G))$ is the multiplier algebra of $C_0(G)\otimes C_b(G)$)

39
votes

2
answers

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### Why are Schur multipliers of finite simple groups so small?

Given a finite simple group $G$, we can consider the quasisimple extensions $\tilde G$ of $G$, that is to say central extensions which remain perfect. Some basic group cohomology (based on the ...

11
votes

1
answer

493
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### Schur multipliers over non-algebraically closed ground fields?

Recently some arithmetic dynamicists came to town, bringing with them some interesting problems in arithmetic geometry.
I started thinking a bit about one of their problems, and it got me wondering ...