Is S_6 the automorphism group of a group? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:27:22Z http://mathoverflow.net/feeds/question/1237 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/1237/is-s-6-the-automorphism-group-of-a-group Is S_6 the automorphism group of a group? Benoit Jubin 2009-10-19T16:59:49Z 2009-11-01T08:16:01Z <p>The automorphism group of the symmetric group S_n is S_n when n is not 2 or 6, in which cases it is respectively 1 and the semidirect product of S_6 with the (cyclic) group of order 2. (For this famous outer automorphism, see for instance wikipedia or Baez's thoughts on the number 6.)</p> <p>On the other hand, S_2 is the automorphism group of Z_3, Z_4 and Z_6 (and only those groups among finite groups). Hence my question: is S_6 the automorphism group of a group? of a finite group?</p> http://mathoverflow.net/questions/1237/is-s-6-the-automorphism-group-of-a-group/1245#1245 Answer by Steven Sivek for Is S_6 the automorphism group of a group? Steven Sivek 2009-10-19T17:57:48Z 2009-10-19T17:57:48Z <p>S_6 is not the automorphism group of a finite group. See H.K. Iyer, "On solving the equation Aut(X)=G", Rocky Mountain J. Math. 9 (1979), no. 4, 653--670, available online <a href="http://rmmc.eas.asu.edu/rmj/rmjVOLS/vol9/vol9-4/iyer.pdf" rel="nofollow">here</a>.</p> <p>This paper proves that for any finite group G, there are finitely many finite groups X with Aut(X)=G, and it explicitly solves the equation for some specific values of G. In particular, Theorem 4.4 gives the complete solution for G a symmetric group, and when n=6 there are no such X.</p> http://mathoverflow.net/questions/1237/is-s-6-the-automorphism-group-of-a-group/1254#1254 Answer by David Zureick-Brown for Is S_6 the automorphism group of a group? David Zureick-Brown 2009-10-19T18:20:15Z 2009-10-19T18:20:15Z <p>On the other hand, S_6 is isomorphic to SL_4(F_2), so that it is an automorphism group in another category (other than groups or sets). This automorphism is exhibited by looking at the 2-torsion of the Jacobian of a hyperelliptic curve H of genus 2 (if H is given by y^2 = f(x), with f of degree 6, then 15 non-trivial two torsion points are given [as a Galois module] by differences of roots of f; see the wiki page for Kummer surface). </p> http://mathoverflow.net/questions/1237/is-s-6-the-automorphism-group-of-a-group/1262#1262 Answer by Jim Fowler for Is S_6 the automorphism group of a group? Jim Fowler 2009-10-19T19:12:06Z 2009-10-19T19:12:06Z <p>It's probably worth pointing to</p> <blockquote> <p>Belolipetsky, Mikhail; Lubotzky, Alexander. Finite groups and hyperbolic manifolds. Invent. Math. 162 (2005), no. 3, 459-472. <a href="http://www.ams.org/mathscinet-getitem?mr=2198218" rel="nofollow"> MR2198218</a>.</p> </blockquote> <p>where it is shown that for every finite group <em>G</em>, there's an infinite group Gamma with Out(Gamma) = <em>G</em>.</p> http://mathoverflow.net/questions/1237/is-s-6-the-automorphism-group-of-a-group/3647#3647 Answer by Pasha Zusmanovich for Is S_6 the automorphism group of a group? Pasha Zusmanovich 2009-11-01T08:16:01Z 2009-11-01T08:16:01Z <p>There is a whole array of results, going back to G. Birkhoff at 1930s saying that every group is an automorphism group of some universal algebra (or some universal algebra inside some class). <br /> (This really should be merely a comment to the previous answer, but I am still not reputable enough to leave comments).</p>