10 votes

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

A reference is: D.F. Holt, The calculation of the Schur multiplier of a permutation group. In: Michael D. Atkinson, Edotor, Computational Group Theory (Conference proceedings, Durham, 1982), Academic ...
Derek Holt's user avatar
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10 votes

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

$H_2(M_{10},\mathbb Z)\cong H^2(M_{10},\mathbb C^\times)\cong H^3(M_{10},\mathbb Z) = \oplus_{ p | 720} H^3(M_{10},\mathbb Z)_{(p)}$ with $p\in\lbrace 2,3,5\rbrace$. A $p$-primary component is ...
Chris Gerig's user avatar
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10 votes

Representation of central extension

The answer is no. According to Theorem 1.3 of this paper, if $H$ is a nilpotent group of class $2$ with cyclic commutator subgroup, then the minimal degree $m_\mathsf{f}(H)$ of a faithful complex ...
Derek Holt's user avatar
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6 votes
Accepted

Reference for Schur multiplier identity

Let $G = F/R$ with $F$ free and $H = S/R$. Since everything is happening modulo $[F,R]$, I am just going to work modulo $[F,R]$. Then, by the Hopf formula, $M(G)$ (the Schur Multiplier) is ...
Derek Holt's user avatar
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4 votes
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Convergence of sequence of images of Schur multipliers

By the uniform bound on $\|A^{(N)}\|$ and linearity, SOT convergence follows from the $\ell^2$-norm convergence, for every $i$, of the $i$-th column $A^{(N)} e_i$ to the $i$-th column $A e_i$. This ...
Mikael de la Salle's user avatar
4 votes

Kernel of a double cover of group as stem extension

In SmallGroup(16,3), the derived subgroup has order 2, is central, and its generator is not a square. (See groupprops.subwiki.org/wiki/SmallGroup(16,3)#Subgroups)
verret's user avatar
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4 votes
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Example of a Schur-nontrivial group with no abelian subgroup of the form $H\times H$?

I presume that since you use the definition $H^2\left(G,U(1)\right)$ for the Schur multiplier, you're interested in finite groups? (For infinite groups this is not in general equivalent to the ...
Jeremy Rickard's user avatar
3 votes

Schur covers of affine 2-transitive groups

Here is an approximate answer. I believe that it is substantially correct, but there might be some small exceptions which I have not thought of. I think that all the examples that are subgroups of ${\...
Derek Holt's user avatar
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3 votes
Accepted

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

The answer is always, yes. Note that there are three classes of involutions in the simply connected version of the algebraic group $E_7$: the central involution $a$, an involution $t$ with centralizer ...
David A. Craven's user avatar
3 votes

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

Here is a general remark about whether a central involution $z$ in a finite group $G$ is a square : It is well known, and easy to derive from the orthogonality relations for group characters and ...
Geoff Robinson's user avatar
3 votes

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

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}$For a field $F$ with $|F|>4$ and $|F|\neq9$, the group $H_2(\SL_2(F))$ has a presentation in terms of the symplectic Steinberg symbols; that ...
Linus's user avatar
  • 553
2 votes
Accepted

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

Beyl-Tappe [4] say on p. 119: 5.9 REMARK. The Schur multiplicators of all finite simple groups have been found, often by exhibiting a universal perfect cover (representation group). For the results ...
Francois Ziegler's user avatar
2 votes

Representation of central extension

Here is an elementary proof of a related general fact. Let $H$ be any finite nilpotent group with $H^{\prime} = Z(H)$ cyclic of order $m$. Let $z$ be a generator of $Z(H)$. Then $\langle z \rangle $ ...
Geoff Robinson's user avatar
1 vote

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

If you really search for a book on most of those topics (at least character tables and conjugacy classes) for simple groups with some proofs, the best (and most compact) ones might be those of Gerhard ...
Mare's user avatar
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