# 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 table or a presentation of the Schur multiplier?

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

• Duplicate of this question. The answer is no -- it's an undecidable problem. Jul 30 at 19:53
• Wait... my group is finite Jul 30 at 19:54
• 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! Jul 30 at 19:56
• @Cloudjr The Wikipedia page for the Schur multiplier has several references for this. See e.g. this paper by Ellis & Leonard. Jul 30 at 20:10
• 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. Jul 30 at 21:44

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;
2

• 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. Jul 31 at 0:04