L. R ONYAI, Computing the structure of finite algebras, J. Symbolic Comput. 9 (1990), 355--373.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....1 ; Vm g, m 2, such that (a) V = V 1 Vm , b) dim k V i = n=m for i = 1; m, and (c) jG : Hj = O(m ) for a constant c 1 0. Remark. Note that Theorems 5.1 and 5.2 include an assumption on char k. The polynomial bound on char k enables a call to Ronyai s algorithm [26] for finding invariant subspaces in deterministic polynomial time. We note, however, that the methods of [18] do not require explicit guarantees of irreducibility, so that, for solvable groups, Theorem 5.1 does not require a polynomial bound on char k. We conjecture that this condition can be ....

....of G. The following proposition, adapted from [18, Theorem 6.1] provides a subroutine to handle certain abelian chief factors of G during this recursion. Note that the algorithm establishing this proposition does not require any assumption on char k since it does not rely on Ronyai s result [26]. such that G 2 d , a linear representation : G GL(V ) and normal subgroups N and A of G such that A N , A is cyclic and uniform, N centralizes A, and N=A is elementary abelian, in polynomial time one can perform one of the following. i) Prove that jG : Aj 24n. ii) Find a normal ....

[Article contains additional citation context not shown here]

L. R ONYAI, Computing the structure of finite algebras, J. Symbolic Comput. 9 (1990), 355--373.

....for this purpose is outlined in [CLG] Furthermore, as one can consider these subgroups as O S , for S the class of cyclic p groups, KL90] asserts that the calculation is possible in polynomial time. For the computation of maximal subgroups we use the algorithms of [LMR94] and [HLOR95] R on90] proves (for equivalent algorithms) that these calculations can be done in polynomial time. The remaining steps for the computation of complements are linear algebra which is of polynomial time, provided the factor presentation is of polynomial length, which [KS99] assures. An implementation of ....

Lajos R onyai, Computing the structure of finite algebras, J. Symbolic Comput. 9 (1990), no. 3, 355--373.

....together with multiplication tables can be computed in polynomial time (at least over most of the interesting ground fields) In this paper we assume that algebras and modules are given by structure constants over K. We intend to use methods based on computing structural invariants of algebras [7, 13, 5] to the module problems outlined above. Unfortunately, finding a nontrivial proper left ideal of is already difficult (essentially as hard as factoring integers) even when is a non commutative simple algebra over Q of dimension four [12] Even worse, it is not known in general whether minimal ....

....(free generators of) a maximal free submodule of V . 2 2.2 Finding a single generator In this subsection we return to the general case where V is a module over the (not necessarily semisimple) K algebra and prove Theorem 1. We compute the radical Rad by the methods of [7] number field case) or [13] (finite field case) Using Rad, we can compute RadV = Rad)V . We consider the action of the factor algebra = Rad on V = V=RadV . Let v 2 V be an arbitrary vector and v = v RadV . It is obvious that v = V implies v = V . We claim that converse also holds. The proof relies on the well known ....

[Article contains additional citation context not shown here]

R' onyai, L. Computing the structure of finite algebras. J. Symbolic Computation 9 (1990), 355--373.

....of sets of commutators (e.g. see [FHL] for P14(i) the general case is an immediate consequence. On the other hand, P14(iii) is new even when K = 1 (cf. x6) An algorithm for P15(i) is discussed in x9. Computation of the abelian part of the socle requires an application of R onyai s work [R o]. P15(ii) is implicit in [BLS1] We assume in P16 that Sigma is specified by a, possibly parametrized, list of names of groups. We outline methods for these problems in x9. P16(ii) is actually implicit in [BLS1] given the additional capability in P12. For K = 1, the special case O p (G) has been ....

....Find the elementary abelian p group V generated by all elements of order p in Z(O p (G) Namely, for each generator d of Z(O p (G) take an element d 0 in hdi of order p; then V is generated by these elements d 0 . Clearly, Soc p (G) V . Since V is a vector space over GF (p) we can use [R o] as follows. Each generator of G induces (by conjugation) a linear transformation of V , whose matrix with respect to a basis of V can be found using 3.1 and linear algebra. Let A be the algebra generated by these linear transformations of V . Then the minimal normal subgroups of G lying in V are ....

[Article contains additional citation context not shown here]

L. R'onyai, Computing the structure of finite algebras, to appear in J. Symbolic Computation (1989).

.... : G GL d (q) specified by the matrix images x 1 ; x t of the generators of G, acting on a d dimensional GF(q) module M , a basic problem is to find invariant submodules, or to prove that G acts irreducibly on M . In theory, this can be done in deterministic time polynomial in d log q [92]; in practice, an idea of Parker [90] the so called Meat Axe, is used. We describe the generalization by Holt and Rees [46] Let A be the algebra generated by x 1 ; x t , and let a 2 A. We compute an irreducible factor p(x) of the characteristic polynomial of a, b : p(a) the null space N ....

Lajos R'onyai. Computing the structure of finite algebras. J. Symbolic Comput., 9:355--373, 1990.

....it does not depend on the field F being small. In fact, on the assumption that the field operations within F are performed using look up tables, its performance is virtually independent of the field. We stress that our priority is to achieve practical speed ahead of theoretical efficiency; in [9] it is proved that the problem of reducing a d dimensional module over a finite field GF (q) can be solved in time which is polynomial in d log(q) When M is reducible, our procedure nearly always produces a basis for a proper submodule, together with the matrices for the group generators on the ....

Lajos R'onyai, Computing the Structure of Finite Algebras, J. Symbolic Computation 9 (1990), 355-373.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC