# Christine Mary Hamill's papers

Christine M Hamill published four papers. The first three are all related to the work of her thesis The Finite Primitive Collineation Groups which contain Homologies of Period Two. which she submitted to the University of Cambridge in 1950. These four papers are:

(i) C M Hamill, On a finite group of order 576, Proc. Cambridge Philos. Soc. 44 (1948), 26-36.

(ii) C M Hamill, On a finite group of order 6,531,840, Proc. London Math. Soc. (2) 52 (1951), 401-454.

(iii) C M Hamill, A collineation group of order $2^{13}.3^{5}.5^{2}.7$, Proc. London Math. Soc. (3) 3 (1953), 54-79.

(iv) H D Deas and C M Hamill, A note on the geometry of lattice planes, Acta Cryst. 10 (1957), 541-542.

We give first the background to Christine Hamill's thesis, which is also the background to the first three of her papers, taken from the paper W L Edge, Obituary: Miss C M Hamill, Edinburgh Math. Notes 40 (1956), 22-25.

1. Background to Christine Hamill's research.
W L Edge, Edinburgh Math. Notes 40 (1956), 22-25.

In 1890 H Burkhardt, whose researches were based on lectures by Felix Klein, published the second part of a long memoir on hyperelliptic modular functions, and in it examined the group of trisection of the periods. This group, of order 25920, had been found 20 years before by Camille Jordan; but Burkhardt constructed 5 theta-functions that were linearly transformed by it and so was able to display it as a group $G$ of linear substitutions on 5 variables. He thereupon worked out the complete set of invariants of $G$, finding the one, $J$ say, of lowest order to be a quartic. When the variables which undergo substitution are taken as homogeneous coordinates in [4], $J = 0$ is the equation of a quartic primal. This primal was encountered by Coble, in 1906, who saw that it has 45 nodes, and gave some details of their configuration. Now H F Baker, on a visit to Göttingen to study under Klein, had there met Burkhardt who gave him offprints of his papers; these Baker studied and copiously annotated and when, nearly 50 years on, retirement from his Cambridge chair had brought comparative leisure he set out to describe, without any dependence on theta-functions, the geometry of the 45-nodal primal, calling it Burkhardt's primal. The outcome was the Cambridge tract: A locus with 25920 linear self-transformations published in 1946. These linear self-transformations, or projectivities, include 45 projections (the homologies of period 2 of Miss Hamill's thesis) whose vertices are the nodes of the Burkhardt primal. Todd, who had found in 1936 a representation of this primal on [3], read the proof-sheets of the tract and noticed that the geometry afforded a means of partitioning the 25920 projectivities into 15 classes such that operations conjugate in $G$ necessarily belonged to the same class. All operations in a class have the same period, and equivalent sets of invariant subspaces, but closer scrutiny may be called for to see whether all operations in the class are conjugate or whether the class is a union of different conjugate sets in G. Todd's results tally in every detail with the separation of $G$ into its 20 conjugate sets accomplished by J S Frame in 1936 by different methods. (Frame had represented $G$ as a group of unitary 4-rowed matrices over a Galois Field of 4 marks.) Todd shows incidentally that every operation of $G$ is the product of 5 or fewer projections.

It was at this juncture that Miss Hamill became Dr Todd's pupil, and he set her to analyse certain larger groups wherein the occurrence of projections had been known since 1914. Fortunate she may well have been, but she soon showed in no uncertain fashion that she could exploit the opportunity that fortune gave her. Before the end of June 1948 she had finished in fair copy the paper On a finite group of order 6,531,840 in which, in the space of 50 pages, the separation into 34 conjugate sets of a group of order 6531840 is set out in full detail,

2. C M Hamill's Introduction to: On a finite group of order 576.

Bagnera (1905) shows that when a certain primitive finite group of order 576, $G_{576}$, is represented as a group of collineations in three-dimensional space, it contains twenty-four harmonic inversions with respect to points and planes, and he shows how to calculate the coordinates of the points and planes of the inversions. The purpose of this note is to discuss the configuration of these points, which we shall call vertices, to show that the whole group is generated by the twenty-four harmonic inversions, termed projections after Prof Baker (1946) and Dr J A Todd (1947), and to enumerate the conjugate sets of the group.

3. J S Frame's review of: C M Hamill, On a finite group of order 576.
Mathematical Reviews MR0022847 (9,268a).

The author obtains the sixteen classes of conjugates of a certain primitive finite group $G$ of order 576, by describing $G$ as a group of collineations which leave invariant a configuration of two associated triads of tetrahedra. Each vertex and opposite face are said to be a pole and its polar plane. The corresponding harmonic inversion is called a projection, and the 24 projections thus obtained are shown to generate the group. The projections form two conjugate sets of 12 each, called type II. Each operation of $G$ is the product of at most four projections. Products of projections from two vertices of the same tetrahedron give 9 elements of period 2 forming a single conjugate set called type $V$. Products of two distinct elements of type $V$ give elements of the same type and also 6 other elements of period 2 forming a conjugate set called type $IX$. It might have simplified the derivations in the paper had the author pointed out that these 15 elements of period 2, together with identity, form an invariant subgroup of order 16, leaving each of the six tetrahedra fixed, and that the quotient group is the direct product of two symmetric groups of degree 3, which represent the permutations of the tetrahedra within the two triads. The conjugate sets for the quotient group could then be split into conjugate sets for $G$.

4. J G Semple's review of: C M Hamill, On a finite group of order 6,531,840.
Mathematical Reviews MR0042403 (13,104a).

This paper, submitted for publication in June 1948 and only now appearing in print, contains the large body of fundamental results on which four other papers, already published before its appearance, were based [E M Hartley, A sextic primal in five dimensions (1950); E M Hartley, Two maximal subgroups of a collineation group in five dimensions (1950), J A Todd, The invariants of a finite collineation group in five dimensions (1950); J A Todd, The characters of a collineation group in five dimensions (1950)]. The group G with which all these papers are concerned is a collineation group of $S_{5}$, of order $2^{8}.3^{6}.5.7$, which was first described by Mitchell. It is generated by 126 homologies, here called projections, and its subordinate collineation groups in the primes of the projections are of the type, generated by 45 projections, which was studied in detail by H F Baker [H F Baker, A locus with 25920 linear self-transformations (Cambridge University Press, 1946)].

The main results of the paper are (i) a complete and concise description of the configuration formed by the 126 vertices; (ii) an analysis, remarkable for its simplicity and elegance, of the operations of the group, their classification into 31 types and 34 conjugate sets; and (iii) a similar analysis of the simple sub-group $G'$ of $G$, of index 2, of which the operations are generated by products of an even number of the 126 projections.

As regards the configuration, the numbers of different types of line, plane, solid and prime that fall to be considered are 2, 4, 7 and 7 respectively; all these spaces are enumerated and an incidence table is given. As regards the analysis of operations of G, the author exhibits all of them as products of at most 6 of the projections; most of the 31 types are obtained by systematic examination of such operations as are products of projections from sets of vertices which lie on the various types of lines, planes, solids and primes of the configuration, the three types which then remain being dealt with separately. The numbers of operations of each type, their periods, the character of their powers, the numbers of projections required to generate them, the number of conjugate sets for each type (always 1 or 2) are exhibited in a comprehensive table, and this also gives corresponding information concerning the simple subgroup $G'$.

5. C M Hamill's Introduction to: A collineation group of order $2^{13}.3^{5}.5^{2}.7$.

The finite primitive collineation groups in $[n]$ which contain homologies of period two (collineations of period two which leave fixed each point of a prime and one point not in the prime) were listed by H H Mitchell (H H Mitchell, 1914). The groups fall into families with the property that each member of a family is a subgroup of the group of highest order in the family, or is homomorphic with such a subgroup. Elsewhere [in her thesis] I have considered each of these families, except the one which consists of the symmetric group of degree $(n + 2)$ represented in $[n]$, and have obtained the operations of each group and their division into conjugacy classes. The methods are those first used by J A Todd (J A Todd, 1947) when he studied the group of order 25920 in (H S M Coxeter, 1940) by means of the geometrical properties of the configuration formed by the 45 centres of the homologies. The results for a group of order $2^{8}.3^{6}.5.7$ in (L E Dickson, 1904), the largest member of the family containing Todd's group, have already been published (Hamill, 1951). Only one other family contains groups in space of more than three dimensions; here I consider the group of highest order in this family and obtain its operations and their division into conjugacy classes. Corresponding results are deduced for the other members of the family.

6. J S Frame's review of: C M Hamill, A collineation group of order $2^{13}.3^{5}.5^{2}.7$.

Following the methods of Todd [J A Todd, On a simple group of order 25920 (1947)] the author studies a family of three finite primitive collineation groups ... which contain homologies of period two, and determines the conjugacy classes by a study of certain configurations in projective 7-space obtained from the vertices of the polytope $(PA)_{8}$. These groups are isomorphic with the rotation groups of the semi-regular polytopes $(IA)_{6}, (SA)_{7}$, and $(PA)_{8}$.

7. Abstract of H D Deas and C M Hamill, A note on the geometry of lattice planes.

This note is an attempt to give a careful restatement of a well-known result in lattice geometry, the proof of the converse part of which does not appear to be so well known.

Last Updated December 2021