COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

104 C. Brott and J. Neubiiser
irreducible representations can be found by applying a theorem of Gaschutz
[5]. For this we give two definitions:
(4.1.1) The socZe S(H) of a group His the product of ail minimal normal
subgroups of H;
(4.1.2) a normal subgroup of a group f-I is called monogenic, if it is
generated by one class of elements conjugate in H.
(4.1.3) THEOREM. A finite group H has a faithful irreducible representation
if and only tf its socle S(H) is monogenic.
In the programme this theorem is used in the following way: If Ng is
transformed into Nh by Nx E GIN, then x-‘gxh-l EN. Hence to check if
S(G/N) is monogenic means checking if the subgroup S 4 G with S/N =
S(G/N) is generated by N and one class of elements conjugate in G. In
preparation for this all monogenic normal subgroups Mi, 1 < i < s == r,
of G are determined. Then for all N Q G, N E pi, 0 e i e t -2, where z; =
{G}, the minimal normal subgroups Kj/N of G/N are found by searching
for normal subgroups Kj * N minimal with respect to this property.
S(G/N) is then equal to S/N where S is the product of all these 4. S(G/N)
is monogenic if for some Mi we have S = NM,.
4.2. Induced representations. Let U e G and M be a C&right-module
of dimension s, T be the matrix representation afforded (uniquely up to
equivalence) by M and let y be its character. The tensor product
Ma = M@c&G (4.2.1)
is a CG-right-module called the module induced from M. Let Tc be the
matrix representation afforded by MC, I+’ its character, (@), the restriction
of yG to U. To can be described as follows. Let
G= UglUUgzU... UUg,, t = G:U and gl = 1, (4.2.2)
be a decomposition of G into cosets of U. Then, for a suitable choice of
the basis of MC, T”(g) is a t X t matrix of SXS blocks for all g E G. The block
(i, j) is equal to T*(giggT’) where
T*(x) =
T(x) for xc U
0 otherwise.
If T is one-dimensional, TG is monomial and
Then
%> = T*tgiayl).
(4.2.3)
(4.2.4)
(4.2.5)
As a character is a class function, it is sufficient to calculate (4.2.5) for
representatives g E G of the classes of elements conjugate in G.
Group characters and representations 105
4.3. M-groups.
(4.3.1) A finite group is called an M-group if all its irreducible representations
are monomial.
As each irreducible monomial representation is induced ([2] Cor. 50.4),
all irreducible representations of such groups are obtained as induced
representations. In order to find M-groups among the factor groups we
use a sufficient criterion of Huppert [6]:
(4.3.2) THEOREM. Let H have a soluble normal subgroup K with all Sylow
subgroups abelian such that H/K is supersoluble, then His an M-group.
We also use the following remark from [6] :
(4.3.3) A finite group H has a uniquely determined normal subgroup U(H)
which is minimal with respect to the property that its factor-group is
supersoluble. For K Q G we have U(H/K) = U(H)K/K.
The programme first determines U(G). If G is supersoluble then U(G) = (1).
Otherwise we use another theorem of Huppert [7]:
(4.3.4) THEOREM. AJinite group H is supersoluble if and only ifall its maximal
subgroups have prime index.
So U(G) is found by searching through all layers Xi, i = O(l)t- 1, for the
first normal subgroup such that all maximal subgroups of G containing it
have prime index. For a factor group G/N, which has a faithful irreducible
representation, U(G)N is formed. In order to decide whether G/N is an
M-group U(G)N/N has to be checked for the following properties :
(1)
G;
(b)
(4
1s U(G)N/N soluble?
U(G)N/Nz U(G)/U(G) n N this can be decided in the following way:
If U(G) is soluble, the same holds for U(G)N/N. Solubility of a subgroup
of G is marked in La.
If U(G) is insoluble, but N soluble, U(G)N/N is insoluble.
Only if U(G) and N are both insoluble, the programme has to check
if the derived series of U(G)N terminates below N.
(2) Are ali Sylow subgroups of U(G)N/N abelian?
If all Sylow subgroups of U(G) are marked as abelian in Lt, this is the
case. Otherwise the programme described in 0 3.2 is used to find the Sylow
subgroups of U(G)N/N and it is checked if their commutator subgroups are
contained in N.
As the property of being an M-group is inherited by factor groups, the
.I.,
checking of (1) and (2) is started with 20.
4.4. Induced monomial representations. We now describe how a subgroup
‘. V can be found whose one-dimensional representations yield irreducible
representations of a fixed factor group G/N by the process of induction. The
choice of U is restricted by a theorem of It8 ([2] Cor. 53.18):