I would like to ask for a reference (book, paper ...) for the following nice construction, which I have found as an exercise in some notes of a course by R. Borcherds. For $n=6$ or $7$ (and only in that case), the alternating group $\mathfrak{A}_n$ has a nontrivial central extension by $\mathbb{Z}/3$. The construction in the case $n=6$ is as follows.

Define an oval in $\mathbb{P}^2(\mathbb{F}_4)$ as a set of 6 points which are in general position (i.e. no 3 of them are collinear). It is easy to show that any projective frame of $\mathbb{P}^2(\mathbb{F}_4)$ is contained in a unique oval, and from this that the stabilizer of an oval in $PGL_3(\mathbb{F}_4)$ is isomorphic to $\mathfrak{A}_6$. The inverse image of this $\mathfrak{A}_6$ in $SL_3(\mathbb{F}_4)$ provides the nontrivial extension by $\mathbb{Z}/3$.

Does anyone know a reference for this? and/or an analogous construction for $\mathfrak{A}_7$?

  • $\begingroup$ Any luck with the references from en.wikipedia.org/wiki/Valentiner_group ? I could only find a brief mention on page 23 of the ATLAS. $\endgroup$ – S. Carnahan Jan 14 '14 at 1:39
  • $\begingroup$ No, all I have seen works over $\mathbb{C}$, with explicit (and complicated) matrices. The Wikipedia article itself mentions the construction, perhaps this is an acceptable reference, thanks! $\endgroup$ – abx Jan 14 '14 at 5:58
  • $\begingroup$ The Wikipedia article is not really independent of the course notes, as far as human sources are concerned - see the edit history. $\endgroup$ – S. Carnahan Jan 14 '14 at 6:43
  • $\begingroup$ Not really perfect as it gives a different construction (as a subgroup of $GL(6,\mathbb{C})$), but quite interesting in any case - thanks to you and to Leandro. $\endgroup$ – abx Jan 14 '14 at 7:39

The oval construction for $3\cdot A_6$ can be found on p.110 of

Symmetric Generation of Groups With Applications to Many of the Sporadic Finite Simple Groups by Robert Curtis.

An e-version is here. In fact Curtis does better: he constructs $3\cdot A_6\cdot 2^2$.

If you read on to p.112 and beyond you will also find a construction of $3\cdot A_7$, as you requested.

(It seems like the e-version is freely available on the web. But if you can't get it, contact me and I'll email it to you.)

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  • $\begingroup$ Hi Nick, It looks like the page is taken down. Could you send it to me at upaudel92@gmail.com ? Thanks :) $\endgroup$ – algeom Nov 5 '14 at 1:26

It is not exactly what you are looking for, but it might be useful.

You can find an interesting construction of the triple covering for the alternating group $\mathbb{A}_6$ in Chapter 2, Section 7.3 of Wilson's book:

Wilson, Robert A. The finite simple groups. Graduate Texts in Mathematics, 251. Springer-Verlag London, Ltd., London, 2009. xvi+298 pp. ISBN: 978-1-84800-987-5 MR2562037 (2011e:20018)

Here you have the link (Google books).

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