## 2-Uniform covering groups of elementary Abelian 2-Group

Saleh, Dana

Saleh, Dana

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##### Publication Date

2023-04-24

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Thesis

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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.

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NUI Galway