How large a subset do you need to uniquely determine a 2-cocycle? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T22:31:08Z http://mathoverflow.net/feeds/question/29234 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/29234/how-large-a-subset-do-you-need-to-uniquely-determine-a-2-cocycle How large a subset do you need to uniquely determine a 2-cocycle? Vipul Naik 2010-06-23T13:21:27Z 2010-06-23T13:42:01Z <p>Suppose <em>A</em> and <em>B</em> are abelian groups. I want to find subsets <em>D</em> of $A \times A$ such that any 2-cocycle $c:A \times A \to B$ for the trivial action is uniquely determined by what it does on <em>D</em>. (Equivalently, if a 2-cocycle is trivial on all of <em>D</em> it is the zero cocycle).</p> <p>I'm not looking for a characterization of all such subsets, just some convenient way of describing some such subsets that makes it easier to list all possible 2-cocycles.</p> <p>By the way, the 2-cocycle condition states that, for all $g,h,k \in A$:</p> <p>$$c(g,h + k) + c(h,k) = c(g,h) + c(g + h,k)$$</p> <p>My first thought was to take a generating set <em>S</em> for <em>A</em> and set $D = S \times S$, but this doesn't seem to work. The problem appears to be that, say, something like $c(g,h + k)$ cannot be described purely in terms of $c(g,h)$, $c(g,k)$, and $c(h,k)$.</p> <p>I'm also interested in the corresponding question when we are restricted to the subgroup of the group of 2-cocycles by one or both of the following conditions: (i) it is skew symmetric, i.e., $c(x,y) = -c(y,x)$, and (ii) $c(a,b) = 0$ whenever $a,b$ generate a cyclic subgroup.</p> <p>For simplicity, please feel free to assume, for instance, that both <em>A</em> and B are <em>p</em>-groups for the same prime <em>p</em>, or even that they are both vector spaces over the field of <em>p</em> elements. My actual reason for asking this question involves trying to compute 2-cocycles in the case $p = 2$, but I thought this question might be of general interest.</p> http://mathoverflow.net/questions/29234/how-large-a-subset-do-you-need-to-uniquely-determine-a-2-cocycle/29236#29236 Answer by Jack Schmidt for How large a subset do you need to uniquely determine a 2-cocycle? Jack Schmidt 2010-06-23T13:42:01Z 2010-06-23T13:42:01Z <p>See section 8.7.2 p307ff of Holt, Eick, O'Brien's Handbook of Computational Group Theory. Also see TwoCohomology in the GAP manual.</p> <p>Roughly speaking normalized cocycles are determined by their values on rule overlaps, and those values are independent of how you overlap. So for instance if A was cyclic of order 8 with generator a, then ζ( a^i, a^(8-i) ) has a constant value for each 0 &lt; i &lt; 8, namely the element b of B such that in your extension G, the image α of a satisfies α^8 = b. It is important to normalize the cocycles, otherwise the formulas get messy. In other words, in the extension G you take a sane transversal of B consisting of reduced words, so there is no loss of generality, just encouragement to keep good hygiene.</p> <p>If A is a general finite group, then you can replace the confluent polycyclic rewriting system by any confluent rewriting system and still have a very effective algorithm. You can also replace it by a general finite presentation, but then determining values of the cocycle become "word problems" in a finitely presented group, and are not very suited to algorithmic determination (though as long as A and B are finite, the problems are only practical).</p> <p>Roughly 1-cohomology corresponds to the bridge between generators and relations, 2-cohomology between relations and associativity (known as "overlaps" when the relations are a rewrite system).</p>