Eamonn O'Brien | |
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 O'Brien receiving the Hector Medal in 2020 | |
| Born | Eamonn Anthony O'Brien |
| Alma mater | Australian National University |
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| Scientific career | |
| Fields | Mathematician |
| Institutions | University of Auckland |
| Thesis | The Groups of Order Dividing 256 (1988) |
| Doctoral advisor | Michael F. Newman |
Eamonn Anthony O'Brien FRSNZ is a professor of mathematics at the University of Auckland, New Zealand, known for his work in computational group theory and p-groups.
O'Brien obtained his B.Sc. (Hons) from the National University of Ireland (Galway) in 1983. He completed his Ph.D. in 1988 at the Australian National University. His dissertation, The Groups of Order Dividing 256, was supervised by Michael F. Newman. [1]
O'Brien's early work concerned classification, up to isomorphism, of groups of order 256. [2] He developed early computer software to complete the classification, and to verify that the classification can correct errors in earlier counting. This led to classifications of many further families of small order groups. In 2000, together with Bettina Eick and Hans Ulrich Besche, O'Brien classified all groups of order at most 2000, excluding those of order 1024. The groups of order 1024 were instead enumerated. [3] This classification is known as the Small Groups Library. Later with Michael F. Newman and Michael Vaughan-Lee O'Brien extended the classifications of groups of order , , and . These classifications comprise the tables provided in the computer algebra systems SageMath, GAP, and Magma.
For a 20-year span from the mid-1990s, O'Brien led the so-called Matrix Group Recognition Project whose primary objective is to solve the following problem: given a list of invertible matrices over a finite field, determine the composition series of the group. [4] [5] Implementations of algorithms that realize the goals of this project form the bedrock of matrix group computations in the computer algebra system Magma.
O'Brien's collaborations include resolution of several conjectures include the Ore conjecture, according to which all elements of non-abelian finite simple groups are commutators. [6]
In mathematics, a group is a set equipped with an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. These three conditions, called group axioms, hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and its definition through the group axioms was elaborated for handling, in a unified way, essential structural properties of entities of very different mathematical nature. Because of the ubiquity of groups in numerous areas, some authors consider them as a central organizing principle of contemporary mathematics.
In mathematics, specifically group theory, given a prime number p, a p-group is a group in which the order of every element is a power of p. That is, for each element g of a p-group G, there exists a nonnegative integer n such that the product of pn copies of g, and not fewer, is equal to the identity element. The orders of different elements may be different powers of p.
In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order
246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71
= 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
≈ 8×1053.
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the conjugating element. They can be realized via simple operations from within the group itself, hence the adjective "inner". These inner automorphisms form a subgroup of the automorphism group, and the quotient of the automorphism group by this subgroup is the definition of the outer automorphism group.
In group theory, the normal closure of a subset of a group is the smallest normal subgroup of containing
In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients. In symbols, a perfect group is one such that G(1) = G, or equivalently one such that Gab = {1}.
In mathematics, computational group theory is the study of groups by means of computers. It is concerned with designing and analysing algorithms and data structures to compute information about groups. The subject has attracted interest because for many interesting groups (including most of the sporadic groups) it is impractical to perform calculations by hand.
1024 is the natural number following 1023 and preceding 1025.
Robert Arnott Wilson is a retired mathematician in London, England, who is best known for his work on classifying the maximal subgroups of finite simple groups and for the work in the Monster group. He is also an accomplished violin, viola and piano player, having played as the principal viola in the Sinfonia of Birmingham. Due to a damaged finger, he now principally plays the kora.
In mathematics, the Nagata conjecture on curves, named after Masayoshi Nagata, governs the minimal degree required for a plane algebraic curve to pass through a collection of very general points with prescribed multiplicities.
The following tables list the computational complexity of various algorithms for common mathematical operations.
In mathematics, the coclass of a finite p-group of order pn is n − c, where c is the class.
Charles R. Leedham-Green is a retired professor of mathematics at Queen Mary, University of London, known for his work in group theory. He completed his DPhil at the University of Oxford.
Aner Shalev is a professor at the Einstein Institute of Mathematics at the Hebrew University of Jerusalem, and a writer.
In mathematics, specifically group theory, a descendant tree is a hierarchical structure that visualizes parent-descendant relations between isomorphism classes of finite groups of prime power order , for a fixed prime number and varying integer exponents . Such groups are briefly called finitep-groups. The vertices of a descendant tree are isomorphism classes of finite p-groups.
In computational group theory, a black box group is a group G whose elements are encoded by bit strings of length N, and group operations are performed by an oracle. These operations include:
Pham Huu Tiep is a Vietnamese American mathematician specializing in group theory and representation theory. He is currently a Joshua Barlaz Distinguished Professor of Mathematics at Rutgers University.
Martin Liebeck is a Professor of Pure Mathematics at Imperial College London whose research interests include group theory and algebraic combinatorics.
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