2-Uniform covering groups of elementary Abelian 2-Group
Date
2023-04-24Author
Saleh, Dana
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Abstract
A covering group of an elementary abelian group of order p(n) is a group G of order
pn+(n
2 ) consisting of the following data:
• G has generators x1, . . . , xn.
• The commutator subgroup of G is equal to the centre and is an elementary abelian
group of order p(n
2 ) or rank
n
2
generated by
n
2
simple commutators [xi, xj].
• G=Z(G) is an elementary abelian group of order p(n), generated by ¯x1, . . . , ¯xn,
where ¯x denotes the coset xZ(G) of Z(G) in G.
In general, an elementary abelian group has many non-isomorphic covering groups
whose enumeration and/or classification is a difficult problem. Different covering
groups are determined by specifying the pth powers of the generators ¯xi as elements
of the elementary abelian group G0. For an odd prime p, the problem can be expressed
purely in terms of linear algebra, because the mapping from G to G’ that takes every
element to its pth power is a linear transformation of Fp-vector spaces, from G=G0 to
G0. For p = 2, this is not the case, and the subject has more of a combinatorial flavour.
An invariant of covering groups of Cn2 is the minimum number k of distinct squares
of elements in a generating set. If k = 1, the corresponding covering groups are called
uniform and it is known that their isomorphism types are in bijective correspondence
with the isomorphism types of simple undirected graphs on n vertices. The goal of
this thesis is to extend this graph correspondence to the case k = 2, which is called
2-uniform. Graphs that encode 2-uniform covering groups are equipped with vertex
and edge colourings, both with two colours. We again obtain a correspondence between
group and graph isomorphism types. Theorem 3.12 presents a class of graphs
that includes at least one representative of every isomorphism type of covering groups.
Many groups are represented by a single graph in this class, and the exceptions are explored
in Chapters 4 through 8. For a 2-uniform covering group G of Cn2 , the uniform
rank (G) of G is defined as the maximum number of elements with the same square in
a minimal generating set, and the uniform corank is n - (G). Theorem 3.8 establishes that covering groups whose uniform corank is at least 4 are almost always represented
by exactly one graph in the class described in Theorem 3.12. The exceptions to
this are investigated in Chapter 4, and the main results are documented in Theorems
4.6, 4.8 and 4.10. Further failures of bijectivity in the correspondence between group
isomorphism types and the graphs of Theorem 3.12 occur for all covering groups of
uniform corank 1, some of uniform corank 2 or 3, and some whose uniform rank is at
most 3. These cases are explored in Chapter 5 (on groups of corank 3), Chapter 6 (on
corank 2), and Chapter 7 (on corank 1). Chapter 7 presents a refinement of the correspondence
of Theorem 3.12, for the special case of covering groups of uniform corank
1. Finally, groups whose uniform rank is at most 3 are considered in Chapter 8.