COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

102 C. Brott and J. Neubiiser
(2.1) (a) as permutations,
(b) as matrices over the rational field R,
(c) as matrices over Z,,
(d) as matrices over R(5) where c is a fixed root of unity,
(e) as three-dimensional affine transformations considered modulo
certain translations,
(f) for soluble groups as abstract generators with a special kind of
defining relations [S].
From these generators the programme @ determines G. For all subgroups
of G their characteristic numbers (as described in [3]) are stored:
all cyclic subgroups of prime-power order are numbered Z1, . . ., Z,.
Then each subgroup U < G is uniquely represented by the n-bit binary
number k(U) whose ith bit is 1 if and only if Zi < U.
Some of the lists obtained by @ are preserved for further calculations.
The programme Xmakes use of a list L1 of all elements in the same form
as the given generators, a list L, of all subgroups, and a list L3 of all
classes of subgroups conjugate in G.
LB contains :
(2.2) for each subgroup U < G:
(a) the characteristic number k(U),
(b) the number of an element in LI which transformes U into a
fixed representative D of the class of subgroups conjugate to
Uin G;
(2.3) for each cyclic subgroup U == G the number in L1 of a fixed generating
element u E U.
L3 contains :
(2.4) for each subgroup fl chosen as representative of its class of conjugate
subgroups :
(a) its order 101,
(b) the number in L3 of its normalizer No(O),
(c) the number in L3 of its centralizer Co(U),
(d) one bit each to characterize if i7 is cyclic, abelian, nilpotent,
supersoluble, soluble, perfect, normal, subnormal or selfnormalizing.
Of the characteristic subgroups determined by @ only G’ is needed by X.
Only for the use in X two further lists are computed. They contain:
(2.5) for each class Ci, 1 4 i =z r,
(a) the number of a fixed representative gi in LI,
(b) the number of elements in Ci,
(c) the order of the elements in Ci;
(2.6) for each element g E G the number of the class containing g.
The class of all elements conjugate to g E G is obtained by transforming g
with representatives of the cosets of Co(g) = C&(g)) in G. Since x-lg’x =
(x-lgxy, it is sufficient to do this for the elements chosen in (2.3).
Group characters and representations 103
3. One-dimensional representations.
3.1. For a one-dimensional representation z of G we have G’ < ker z.
Hence z induces a mapping z’ : G/G’-C given by
z’(G’g) = z(g) for all g E G. (3.1.1)
Let H = G/G’ be of order 1 H 1 = p? . . . p;~; then H is the direct product
of its Sylow subgroups S,,, . . . , S,, of orders pff, 1 == i < t. S,, is a direct
product of cyclic groups of orders pf”, . . . , pf’sf with eir+ . . . +eq = ei.
Let m = slf . . . + s, and let x1, . . . , X, be generators of the cyclic
direct factors of H thus obtained. For each z’ the values z’(xJ form an
m-tuple (wl, . . ., w,) with wi E C and wpl = 1. Conversely all the 1 H 1
such m-tuples are different one-dimensional representations of H in C.
3.2. Let G’ be in the layer (cf. [3]) C,, let Sz/G’ be the pi-Sylow subgroup
of G/G’. Then Sz is found in Eeerfr by searching for a subgroup which
contains G’ and is of order p;*lG’I. The programme finds a decomposition
of S,*i/G’ into cyclic direct factors in the following way :
If Sg/G’ is not cyclic, a normal subgroup U with G’ < U e Sic and
U/G’ cyclic is searched for in E,.+l which has an Sz-complement V
module G’ in .Xr+ei-l. If no such normal subgroup U is found in -&+,
the search for U is continued in z,,, with V in zr+er+
If U and V have been found, U is stored away as a direct factor of
Sg/G’, Sz is replaced by V and the same process is continued. If G is
abelian, L3 yields the information whether U = U/G’ is cyclic, otherwise
we use that U/G’ is a cyclic p-group if and only if there is exactly one
maximal subgroup of U containing G’. All these decisions require only
calculations with characteristic numbers, so this part of X is very fast.
3.3. Let F be a fixed primitive nth root of unity. Let the classes of a cyclic
group U = (x) of order n be Ci = {xi-l}; then the one-dimensional
representations can be numbered in such a way that
Zj(Ci) = @-lw-l) , 1 ==i;J