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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 table or a presentation of the Schur multiplier?

If possible please refer me to a paper which talks about the algorithm.

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  • $\begingroup$ Duplicate of this question. The answer is no -- it's an undecidable problem. $\endgroup$ Jul 30 at 19:53
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    $\begingroup$ Wait... my group is finite $\endgroup$
    – Cloud jr
    Jul 30 at 19:54
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    $\begingroup$ The case of when $G$ is a finite group is answered in the comments of that question, too. Those comments link you here for explicit algorithms (e.g. in GAP). Welcome to MathOverflow, by the way! $\endgroup$ Jul 30 at 19:56
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    $\begingroup$ @Cloudjr The Wikipedia page for the Schur multiplier has several references for this. See e.g. this paper by Ellis & Leonard. $\endgroup$ Jul 30 at 20:10
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    $\begingroup$ I don't have access to it right now, but I believe that the following book explains algorithms for this (and many other things -- if you have computational questions in group theory, it's a good first place to look): Holt, Derek F.; Eick, Bettina; O'Brien, Eamonn A. Handbook of computational group theory. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2005. $\endgroup$ Jul 30 at 21:44
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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 Press, 1984, pages 307-319.

But you might have difficulty finding it!

The idea of this algorithm is to find the $p$-parts of the multiplier separately for the primes $p$ dividing $|G|$. To do that, we first find the multiplier $M(P)$ of a Sylow $p$-subgroup $P$ of $G$ using part of the $p$-quotient algorithm, and then find the $M(G)_p$ as the subgroup of $G$-stable elements of $G$. (I programmed this myself first in ALGOL60 and then in C in the early 1980s, partly motivated by the fact that there had been so many errors in the calculation of the multipliers of the finite simple groups - it had taken three attempts to get $M(M_{22})$ right!)

There is a much simpler algorithm available in Magma as $\mathtt{Darstellungsgruppe}$ that takes as input a finite presentation $\langle X \mid R \rangle$ of the finite group $G$, and finds a presentation of a Schur-covering group $C(G)$ of $G$ by factoring out a free abelian subgroup of $R/[F(X),R]$ in $F(X)/[F(X),R]$ using the Hopf formula. The multipler can then be calculated as the kernel of the natural map $C(G) \to G$. This works OK but only for moderately small groups $G$. Here is an example with $G=A_5$.

> G:=Group<x,y|x^2,y^3,(x*y)^5>;
> C,phi:=Darstellungsgruppe(G);
> K:=Kernel(phi);
> #K;
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    $\begingroup$ For the interested, the proceedings (and thus the article) can be found by searching for "Computational group theory: Proc. London MS symposium" in whatever pseudo-legal book repository one knows of. $\endgroup$ Jul 31 at 0:04

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