Table of Contents
What is the monster in group theory?
The monster can be realized as a Galois group over the rational numbers, and as a Hurwitz group. The monster is unusual among simple groups in that there is no known easy way to represent its elements. This is not due so much to its size as to the absence of “small” representations.
What does Galois theory state?
The central idea of Galois’ theory is to consider permutations (or rearrangements) of the roots such that any algebraic equation satisfied by the roots is still satisfied after the roots have been permuted. Originally, the theory had been developed for algebraic equations whose coefficients are rational numbers.
How was the monster group found?
The first of the sporadic simple groups was discovered in the nineteenth century by French mathematician Émile Mathieu. It wasn’t until 1973 that two mathematicians—Bob Griess at the University of Michigan and Bernd Fischer at Universität Bielefeld—independently predicted the existence of the monster.
Is Galois group Abelian?
. So the Galois group in this case is the symmetric group on three letters, which is non-Abelian.
How many elements does the monster have?
It contains the following number of elements: 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 (more than the number of quarks in the Sun).
What number is the monster?
The least number of dimensions in which the Monster group can act non-trivially is 196,883. This number is 47.59. 71, the product of the three largest prime numbers dividing the size of the Monster, but its main point of interest is that by adding 1 we obtain 196,884.
What is the order of Galois group?
The order of the Galois group equals the degree of a normal extension. Moreover, there is a 1–1 correspondence between subfields F ⊂ K ⊂ E and subgroups of H ⊂ G, the Galois group of E over F. To a subgroup H is associated the field k = {x ∈ E : f(x) = x for all f ∈ K}.
What is a Galois group in math?
In mathematics, more specifically in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called…
What is the Galois group of the polynomial x2 – 4x + 1?
We conclude that the Galois group of the polynomial x 2 − 4x + 1 consists of two permutations: the identity permutation which leaves A and B untouched, and the transposition permutation which exchanges A and B. It is a cyclic group of order two, and therefore isomorphic to Z/2Z.
What is the Galois theory of automorphism?
The theory has been popularized among mathematicians and developed by Richard Dedekind, Leopold Kronecker, Emil Artin, and others who in particular interpreted the permutation group of the roots as the automorphism group of a field extension. Galois theory has been generalized to Galois connections and Grothendieck’s Galois theory.
Is the Galois group modulo a prime isomorphic to a prime?
It is known that a Galois group modulo a prime is isomorphic to a subgroup of the Galois group over the rationals. A permutation group on 5 objects with elements of orders 6 and 5 must be the symmetric group S 5, which is therefore the Galois group of f(x).