How large a subset do you need to uniquely determine a 2-cocycle? - MathOverflow

How large a subset do you need to uniquely determine a 2-cocycle?

2010-06-23T13:21:27Z

Suppose A and B are abelian groups. I want to find subsets D 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 D. (Equivalently, if a 2-cocycle is trivial on all of D it is the zero cocycle).

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.

By the way, the 2-cocycle condition states that, for all $g,h,k \in A$:

$$c(g,h + k) + c(h,k) = c(g,h) + c(g + h,k)$$

My first thought was to take a generating set S for A 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 $c(h,k)$.

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.

For simplicity, please feel free to assume, for instance, that both A and B are p-groups for the same prime p, or even that they are both vector spaces over the field of p 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.

Answer by Jack Schmidt for How large a subset do you need to uniquely determine a 2-cocycle?

2010-06-23T13:42:01Z

See section 8.7.2 p307ff of Holt, Eick, O'Brien's Handbook of Computational Group Theory. Also see TwoCohomology in the GAP manual.

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 < i < 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.

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).

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).