Questions tagged [schur-multipliers]

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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 ...
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9 votes
1 answer
457 views

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 $S_6$, ...
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7 votes
2 answers
434 views

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)}(\...
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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 ...
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1 vote
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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 ...
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  • 261
2 votes
0 answers
61 views

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\...
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  • 261
2 votes
2 answers
208 views

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 ...
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4 votes
2 answers
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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 ...
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  • 261
9 votes
1 answer
270 views

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{\...
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6 votes
0 answers
159 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^...
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5 votes
1 answer
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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 ...
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2 votes
1 answer
129 views

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 ...
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2 votes
1 answer
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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 $1 \to N \to G \to X \to 1$. ...
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4 votes
0 answers
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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 ...
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1 vote
1 answer
175 views

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 ...
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1 vote
0 answers
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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 ...
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2 votes
1 answer
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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 \...
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4 votes
0 answers
170 views

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 ...
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7 votes
1 answer
410 views

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 ...
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1 vote
1 answer
326 views

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/...
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1 vote
0 answers
97 views

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 [...
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13 votes
2 answers
710 views

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 (...
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0 votes
0 answers
143 views

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 $C_0(G)\otimes C_b(G)$)
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36 votes
2 answers
3k views

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 ...
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11 votes
1 answer
464 views

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 ...
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