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The 5th roots of unity in the complex plane form a group under multiplication. Each non-identity element generates the group. In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses.
The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a concise form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on any representative of the respective conjugacy class of G (because characters are class functions).
A presentation of a group determines a geometry, in the sense of geometric group theory: one has the Cayley graph, which has a metric, called the word metric. These are also two resulting orders, the weak order and the Bruhat order, and corresponding Hasse diagrams. An important example is in the Coxeter groups.
In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations. It is much used in geometric topology , the fundamental group of a simplicial complex having in a natural and geometric way such a presentation.
The automorphism group of the free group with ordered basis [ x 1, …, x n] is generated by the following 4 elementary Nielsen transformations: Switch x 1 and x 2; Cyclically permute x 1, x 2, …, x n, to x 2, …, x n, x 1. Replace x 1 with x 1 −1; Replace x 1 with x 1 ·x 2; These transformations are the analogues of the elementary row ...
Later Dunfield and Thurston proved [32] that if a one-relator two-generator group = , = is chosen "at random" (that is, a cyclically reduced word r of length n in (,) is chosen uniformly at random) then the probability that a homomorphism from G onto with a finitely generated kernel exists satisfies
The group Sz(2) is solvable and is the Frobenius group of order 20. The Suzuki groups Sz(q) have orders q 2 (q 2 +1)(q−1). These groups have orders divisible by 5, but not by 3. The Schur multiplier is trivial for n>1, Klein 4-group for n=1, i. e. Sz(8). The outer automorphism group is cyclic of order 2n+1, given by automorphisms of the field ...
The free abelian group on S can be explicitly identified as the free group F(S) modulo the subgroup generated by its commutators, [F(S), F(S)], i.e. its abelianisation. In other words, the free abelian group on S is the set of words that are distinguished only up to the order of letters.