GUÍA PARA EL DISEÑO DEL MANUAL DE PROCEDIMIENTOS

A . Group Tho o ry

The subsection deals w ith a negative re s u lt about applications of the algorithms fo r permutation groups on abstract groups. The gap between the upper bound of 0 ( | G | ^ +e) on the time required fo r

computations ( lik e order, canonical structure of a group) in abstract abelian groups represented by a set of generators and the upper bound of 0 (lo g c |G|) fo r some constant on the time required f o r s im ila r computations in abelian permutation groups is fa r too b ig . As a bridge between permutation groups and abstract groups is C a yle y's theorem (see H all [2 2 ]) saying that every group is isomorphic to a permutation group.

I t w ill be shown that given a set of generators fo r a group G, the time required by an optimal algorithm fo r computing the isomorphic image of G Into a permutation group 1s n (| G | ). Moreover, C ayle y's construction is shown to be optimal w ith in a polynomial fa cto r.

Suppose that G = <g> c y c lic elementary p-group f o r p prime. Then there e x ists a permutation w such that <ir> = G. Then the order o f ir is |G| - p which Implies that it permutes a t least |G| symbols. Therefore an optimal algorithm fo r computing an isomorphic permutation group to G requires fl(|G|) elementary operations 1n the w orst-case.

Moreover Cayley's construction of the Isomorphic group requires |G| elementary operations, given G ■ < g ^ , . . . , g j c > ,T h e construction 1s the fo llo w in g : Define a permutation r g^ fo r 1 < 1 < n acting on {g :g € G> such that r g^ (g ) ■ gg^ v g € G. Then 1t 1s not d i f f i c u l t to show that

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I t 1s not d i f f i c u l t to see that the computation of r g requires 0(|G |) and then the isomorphic image of G requires 0(k|G|£) elementary operations. Assuming tha t k w ^ m 0 (1 ogc |G|) fo r some constant c > 0, Cayley's method

1s optimal w ithin a fa cto r of 0(1ogc |G|).

Moreover 1n the case which G 1s abelian one can construct a

complete basis « b j , . . . , b n» fo r 6 1n 0( |G|1 ^2",’eC) elementary operations. I f c^ are d is jo in t cycles permuting elements 1n { 1 , . . . , |G|}and c. 1s of length |b^| fo r 1 < 1 < n, then

^C| i... > c ^ w G.

Th is computation requires 0(|G|) elementary operations and thus 1s optim al.

Therefore one can conclude that the representation of the group as permutation groups 1s powerful fo r computing the order of the group but the computation of an Isomorphic permutation group to a given abstract group is in tra c ta b le .

B. Graph Theory

The following two problems are polynomial time equivalent:

Group Intersection problem: Given two permutation groups, subgroups of the symmetric group Sn , compute a generating set fo r th e ir In te rse ctio n . Graph Isomorphism Problem: Given two graphs determine whether or not they are Isomorphic and 1f so, construct an Isomorphic from the one to another.

There Is no known polynomial time algorithm fo r both o f the problems mentioned above. Hoffman In [2 4 ] suggests that th e ir complexity lie s between P and NP and th a t they do not seem to be candidates of the

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I t 1s not d i f f i c u l t to see that the computation o f requires 0(|G|) and then the Isomorphic Image of G requires 0(k|G|€) elementary operations. Assuming that k m g m 0(1 ogc |G| ) fo r some constant c > 0, Cayley's method 1s optimal w ith in a fa cto r o f 0 (lo g c |G|).

Moreover 1n the case which G 1s abelian one can construct a

complete basis « b 1 t. . . , b n» fo r G 1n 0 (| G | ^ 2+eC ) elementary operations. I f c^ are d is jo in t cycles permuting elements 1n { 1 , . . . , |G|>and c^ 1s of length |b^| fo r 1 < 1 < n, then

<c ^ , . « « |C^^ w G.

This computation requires 0(|G|) elementary operations and thus 1s optim al.

Therefore one can conclude that the representation of the group as permutation groups Is powerful fo r computing the order of the group but the computation of an Isomorphic permutation group to a given abstract group 1s In tra cta b le .

B. Graph Th e o ry

The fo llo w ing two problems are polynomial time equivalent:

Group In tersection problem: Given two permutation groups, subgroups o f the symmetric group Sn , compute a generating set fo r th e ir In te rse ctio n . Graph Isomorphism Problem: Given two graphs determine whether or not they are Isomorphic and 1f so, construct an Isomorphic from the one to another.

There 1s no known polynomial time a lgo rith m fo r both of the problems mentioned above. Hoffman In [2 4 ] suggests th a t th e ir complexity lie s between P and NP and th a t they do not seem to be candidates of the

NP-complete class o f problems (f o r d e fin itio n s o f P, NP and NP-complete, see Aho e t a l. [ 2 ] ) .

The graph isomorphism problem has been considered under constraints e .g . the graph Is of bounded valence (see Luks [ 3 8 ] ) . The algorithms described in the previous sections of th is chapter have no d ire c t a p p lic a tio n on the graph Isomorphism; they (algorithm 4.12.?) may help fo r the solution o f the graph isomorphism problem under the constraint that the automorphism group of the graph is isomorphic.

C . Chem istry

I t 1s well-known the connection o f the representation of the molecules and the symmetric group (see [ 3 ] ) . Many of these

representations form an abelian group (see [1 1 ], e .g . tra nsla tio n group). The computation o f the stru ctu re of these groups aids to the computation of the o rb ita ls o f the atoms (see [ 3 ] ) .

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