COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

106 C. Brott and J. Neubiiser Group characters and representations 107
(4.4.1) THEOREM. The dimension of an irreducible representation of a group
H is a divisor of the index of each abelian normal subgroup A of H.
We call the greatest common divisor of the indices of the maximal abelian
normal subgroups of H the It&index i(H). By (4.4.1) we have to find a subgroup
U== N such that G: U 1 i(G/N). In order to determine i(G/N) we search
for Kj a G with N a Ki, K,/N abelian, and maximal with respect to these
properties. Then i(G/N) is calculated from the orders of the Kt listed in LI.
We also use 2 d,? = 1 G I. As there are IG/G’I one-dimensional represen-
i=l
tations, only r - IG/G’I representations of dimensions a 2 are missing. For
their dimensions we have:
i:
i=p/~'l+i
d;= IGI-IG/G’I.
Hence2eG: U< 1/lGl- IG/G’I -4(r- IG/G’I - 1) is a restriction for
the index of U.
Finally we can restrict the search for U to representatives of the classes
of subgroups conjugate under G, as conjugate subgroups yield equivalent
representations.
The kernel V a U of the one-dimensional representation T of U used for
the induction process must satisfy the requirements
U/V cyclic, (4.4.2)
N = ker TG = n Vg. (4.4.3)
HG
The programme X deals with all factor groups G/N with faithful irreducible
representations in turn. If G/N cannot be recognized as an M-group
by (4.3.2), the user of the programme is informed that possibly not all
irreducible representations of G/N will be found. In spite of this, the programme
tries to find irreducible monomial representations of G/N.
First, the programme searches for subgroups U c Es,, s < t- 1, such that
N a U and G : U is a dimension allowed by the restrictions. Then subgroups
VC &, i< s- 1, V a U are searched for, which satisfy (4.4.2) and (4.4.3).
This is done in the following way: if V a G then also V a U. Otherwise the
normalizer N,(V) is determined from Lz and LB. If U=S No(V) we have
V a U. U/V is cyclic if and only if for each prime pi j U/V I there is exactly
one maximal subgroup of index p in U, containing V. If V a G we must
have V = N, otherwise the intersection of all conjugates of V in G is formed
and this must be equal to N. If V is found, meeting all requirements, N, U
and V are listed.
For the determination of faithful one-dimensional representations of
U/V we have to find an element x E U with (x, V) = U. This is found as a
generator of a cyclic subgroup 2 of G meeting the following requirements :
2~ U, Z=k V and ZV*/= Mi for all maximal subgroups Mi of U containing
v.
The whole process described in 9 4.4 needs only calculations with characteristic
numbers and hence is rather fast.
4.5. A test for irreducibility. We use the inner product of characters
to check the irreducibility of TG. By a special case of Frobenius’ reciprocity
theorem ([2], 38.8), for any character y of a subgroup U< G we have:
((YG>U, Y.l = (YG, Y3. (4.5.2)
Since a character x of a representation of G is irreducible if and only if
(x, x) = 1, we have:
yG is irreducible if and only zf
((Y3r.h WI = &-p(U)~(U) = 1.
Let e be the exponent of G. Then the values of yG, and hence (UI *((yG)o, y),
are sums of eth roots of unity, i.e. sums of powers of a fixed primitive
eth root of unity, E say.
For the calculation of the sum (4.5.2) we count in a list L the number nj
of times ej, 0 ~j < e - 1, occurs as a summand of I U/ * ((Y~)~, y). We then
use the fact that a sum of nth roots of unity is equal to zero if and only if it
can be decomposed into sums over cosets of nontrivial subgroups of the
group of al lnth roots of unity ([9], p. 240). Because of this theorem the programme
proceeds as follows : for each divisor d of e and each i, 0 =S i =S d- 1,
the smallest of the numbers ni, nd+i, nw+i, . . . is subtracted from all
these numbers. yG is irreducible if and only if, after doing this, no = 1 Ul
andni = Oforalli, lsiee-1.
4.6. The calculation of induced representations. The programme deals in
turn with the triplets N, U, V previously found. First G is decomposed into
cosets of U, then for each faithful one-dimensional representation Ti of
U/V the calculation is performed in four steps:
(1) Ti is calculated in the same way as described in 9 3.3 for the onedimensional
representations of G.
(2) The character-values xl” of the induced representations TjG are calculated
for representatives of the classes of elements conjugate in G.
(3) The irreducibility of Ty is tested by the method described in 0 4.5.
(4) If q passes the test, the values of x$j) are brought into a normal form
by a method analogous to that described in 9 4.5, and are compared
with the list of irreducible characters previously obtained in order to
decide if 7 is a new irreducible representation.
If required the kernel of 7, the matrices representing generators or all