The modular isomorphism problem asks whether FG congFH implies that G congH for two p-groups G and H and F the field with p elements. This problem is still open, despite various efforts towards proving the claim or finding counterexamples to it. The claim has been proved, for example, for abelian p-groups Des56, p-groups of class 2 and exponent p PS72, metacyclic p-groups San96 and groups of order pn dividing 27 citeWur93, BKRW99 or p5 SS96.
The modular isomorphism problem can be considered as a special case of a wider range of problems. In general, one can ask under which conditions on a ring R and groups G and H does RG congRH imply G congH. Important results on this type of problem are due to Dade Dad71 who found that for every field R there exist finite groups G notcongH with RG congRH and to Hertweck Her95 who determined finite groups G notcongH with ZG congZH. In both cases, explicit examples for G and H are known, but none of them are p-groups. On the other hand, Roggenkamp and Scott RSc87 proved that RG congRH implies that G congH if R is the ring of p-adic integers and G and H are p-groups.
Computational approaches have been used to investigate the modular isomorphism problem. Based on an algorithm by Roggenkamp and Scott RS93, Wursthorn Wur93 described an algorithm for checking the modular isomorphism problem; that is, he described an algorithm for checking whether two modular group algebras FG and FH are isomorphic. This algorithm has been implemented in C by Wursthorn and has been used applied to the groups of order dividing 27 without finding a counterexample, see BKRW99.
This package contains an implementation of the new algorithm described in Eic07 for checking isomorphism of modular group algebras. Hence it provides a new method to investigate the modular isomorphism problem. The algorithm underlying this method can also be used to check isomorphisms and to compute the automorphism group of a finite dimensional, nilpotent, associative algebra over a finite field.
Let A be an associative algebra of dimension d over a field F of prime order and let {b1, ..., bd} be a basis for A. We identify the element x1 b1 + ...+ xd bd of A with the element (x1, ..., xd) of Fd and thus obtain an identification of A with the vector space Fd. The multiplication of A can then be described by a structure constants table: a 3-dimensional array with entries ai,j,k inF satisfying that
bi bj = sumk=1d ai,j,k bk.
A canonical form Can(A) for A is a structure constants table for A which is unique for the isomorphism type of A; that is, two algebras A and B are isomorphic if and only if Can(A) = Can(B) holds. Further, using the above identification of A with Fd the automorpism group of A can be defined as
Aut(A) = { M inGL(d,F) mid(ab)M = (aM)(bM) for all a,b inA }.
An associative algebra A is nilpotent if its power series terminates at the trivial ideal of A; that is, the algebra A has the series
A > A2 > ...> An > An+1 = {0} where Aj is the ideal of A generated by all products of length i in A. Note that A is generated by dim(A/A2) elements in this case and A does not contain a multiplicative identity.
Let G be a finite p-group and F the field with p elements. Then FG is the modular group algebra defined by G; it is an associative algebra, contains a multiplicative identity and has dimenion |G|.
Every modular group algebra FG contains a unique maximal ideal J(FG); this ideal has codimension 1 in FG and a basis is given by { g-1 mid g inG, g neq1 }. Thus J(FG) coincides with the augementation ideal of FG. It is well-known that J(FG) is nilpotent and a basis through its power series can be read off from the Jennings series of G.
By construction, the ideal J(FG) is fully invariant under any isomorphism of FG. Hence we find that Aut(FG) can be identified with Aut(J(FG)) and a canonical form Can(J(FG)) extends naturally to a canonical form of FG. In particular, two group algebras FG and FH are isomorphism if and only if J(FG) and J(FH) are isomorphic or, equivalently, if and only if Can(J(FG)) = Can(J(FH)) holds.
The main function of this package determines the automorphism group of J(FG) and a canonical form Can(J(FG)).
The modular isomorphism problem asks whether there exist two non-isomorphic finite p-groups G and H with FG congFH. The methods of this package can be used to check this problem for groups of small order. In fact, the implementation of this package has been used to check MIP for the groups of order dividing 28 and 36 (without finding a counterexample).
Example manual