COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

212 P. G. Ruud and R. Keown
TX where each of the elements of K is either 1 or - 1. The Frattini subgroup
of G is the intersection of the kernelsof all irreducible representations obtained
from such K’s.
Proof It is clear that each K, as defined above, defines a one-dimensional
representation of G, and that distinct pairs, K and K’, define distinct irreducible
representations. Conversely, every one-dimensional representation
T of G determines a unique K. Thus every one-dimensional representation
of a 2-group G can be obtained immediately from its Hall-Senior defining
relations. Each K, containing at least one - 1, whose elements are either
1 or - 1, determines an irreducible representation T one-half of whose
values are 1 and the other one-half are - 1. The kernel of the corresponding
TK is clearly a maximal subgroup M of G. Conversely, each maximal subgroup
A4 generates a one-dimensional representation corresponding to a
K of this type. The Frattini subgroup is the common part of these kernels.
The subgroup (C) generated by C is contained in the derived group G’.
To see that (C)coincides with G’, let bj be any generator coming later in the
sequence of generators than the elements of C. We construct a K to show
that no element of G having b, in its standard expansion is an element of 6’.
Let kj have the value - 1 and let k, have the value 1 for m different fromj
except when bi equals by In this exceptional case, which can occur for at
most one b,, let k, have the value i. Let TK be the corresponding one-dimensional
representation and note that any element of G whose standard expansion
contains bj does not belong to the kernel of TK. It follows that the elements
of C generate G’.
We turn to the calculation of the higher dimensional irreducible representations
of G. Note that a 2-group of order 16 or less cannot have an
irreducible representation of dimension greater than two and that a 2-group
of order 32 or 64 cannot have an irreducible representation of dimension
greater than four. Let the central series of G determined by the Hall-Senior
defining relations be given by
{l}cHlc . . . cH,=G.
Recall that H,, 1~ r == n, denotes the subgroup (bI, . . . , b,) determined by
the first r generators. The irreducible representations of H, can be determined
by induction from the irreducible representations of Hr.-, and its subgroups.
Some, perhaps all, of the two-dimensional representations of Hr
can be calculated immediately by induction from the pairs of conjugate
one-dimensional representations of Hr-,. However, the two-dimensional
self-conjugate representations of Hr-,, if any exist, give rise to a more
troublesome problem. Each of these, say t, is the orbit of a pair, T and T’,
of associated two-dimensional irreducible representations of Hr. The representation
R induced by t is the direct sum of the associated pair, Tand T’.
To avoid the reduction problem, we note that each self-conjugate twodimensional
irreducible representation t of H,-, is an induced representa-
Irreducible representations of finite groups 213
tion from either member of a pair, s and s’, of conjugate representations of
some normal subgroup Kof H,-,. The subgroup Kcan be readily identified
from the representation t since it consists of those elements of H,-, which
are mapped into diagonal matrices by t. Furthermore, the pair, s and s’, of
conjugate representations of Kcan be read off from the entries of these diagonal
matrices. Observe that each of the associated representations, T and
T’, of Hr subduce the representation t on Hr-, and, consequently, give rise
to the same matrices for K as t. Moreover, either of T and T’, say T, must
be an induced monomial representation obtained by induction from a onedimensional
representation of some normal subgroup H, containing K, of
index two in Hr. The set of normal subgroups, containing K, of index two
in H, can be determined immediately from the one-dimensional representations
of H,.. Each of these is of the form (K, g) for an easily determined
element g of Hr. Such a subgroup (K, g) is a suitable H for our purposes
only if the irreducible representations of Kis carried into itself under conjugation
by g, a property easily checked from the data available.
When a suitable H has been determined, one computes the pair, q and r,
of associated one-dimensional representations of H, each of which subduces
the representations on K. The representation q of Hinduces one of the
pair, T and T’, of associated representations of H, each of which has [t] for
its orbit. The representation r induces the other. This completes the construction
of the irreducible, two-dimensional representations of Hr. When
the order of H,. exceeds 16, there may exist four-dimensional irreducible
representations of H,. These arise from conjugate pairs of two-dimensional
or from self-conjugate four-dimensional irreducible representations of Hr-,.
The construction of the self-associated four-dimensional representations of
H, from the conjugate pairs of two-dimensional representations of H,-, is
straightforward. The construction of the associated four-dimensional representations,
T and T’, of H, from a self-conjugate four-dimensional irreducible
representation t of H,-, is very much as before. Select the subgroup
K of index four in H,-, whose elements correspond to diagonal matrices
under the representation t. A set [sr, SZ, s3, s4] of four mutually conjugate
representations of K can be determined from the diagonal matrices which
are images of K under the representation t. The subgroups of index four of
H, which contain K are of the form (K, g) and can be determined from the
available data. Such a subgroup H of Hr is suitable for our purposes only if
the representation s1 of K is carried into itself by conjugation under g. This
being the case, s1 is a self-conjugate representation of Kconsidered as a subgroup
of index two in H. Consequently, si induces a pair, q and r, of associated
one-dimensional representations of H. The representation q of H
induces one of the associates with orbit t and the representation r induces
the other.
This concludes the outline of the method of calculation. The next section
is concerned with the programs for the calculation.