Reconstruction Conjecture: Group theoretic formulation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T14:04:42Z http://mathoverflow.net/feeds/question/34914 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/34914/reconstruction-conjecture-group-theoretic-formulation Reconstruction Conjecture: Group theoretic formulation To be cont'd 2010-08-08T11:20:06Z 2011-02-05T23:49:27Z <p>As we read from wiki, informally, the <a href="http://en.wikipedia.org/wiki/Reconstruction_conjecture" rel="nofollow">reconstruction conjecture</a> in graph theory says that graphs are determined uniquely by their subgraphs.</p> <ol> <li>Is there a group-theoretic formulation of this conjecture?</li> <li>Has an analogous conjecture been made in group theory(in any sensible way)?</li> </ol> <p>References such as books will do, thanks. </p> http://mathoverflow.net/questions/34914/reconstruction-conjecture-group-theoretic-formulation/54476#54476 Answer by Mark Sapir for Reconstruction Conjecture: Group theoretic formulation Mark Sapir 2011-02-05T23:49:27Z 2011-02-05T23:49:27Z <p>There are continuum 2-generated groups where all proper subgroups are cyclic of order \$p\$ (for the same prime \$p\sim 10^{70}\$), the Tarski monsters. All these groups have the same lists of proper subgroups, and the same lattice of subgroups. Also all cyclic groups of prime order have the same proper subgroups: the trivial group. So in general the answer is "no". But every finite non-cyclic abelian group is determined by the list of its proper subgroups, which follows easily from the description of finite abelian groups (note that I and, I think, the question, are not talking about the lattice of subgroups, but about the list of all proper subgroups). I suspect that at least for a large class of finite solvable groups, the list of proper subgroups determines the group. I would look first at A-groups, finite groups with all Sylow subgroups abelian (see the book of Huppert, "Endliche Gruppen") because the structure of subgroups of A-groups is more known, and one can use induction on the order of the group. </p>