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Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure. [1] [2] For an abelian group, each conjugacy class is a set containing one element (singleton set).
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).
Let F 2 11 be the finite field with 2 11 elements. Its group of units has order 2 11 − 1 = 2047 = 23 · 89, so it has a cyclic subgroup C of order 23.. The Mathieu group M 23 can be identified with the group of F 2-linear automorphisms of F 2 11 that stabilize C.
The entries in the same row are in the same conjugacy class. Every entry appears once in each column, as seen in the file below. Every entry appears once in each column, as seen in the file below. The positions of permutations with inversion sets symmetric to each other have positions in the table that are symmetric to each other.
Co 3 acts on the unique 23-dimensional even lattice of determinant 4 with no roots, given by the orthogonal complement of a norm 4 vector of the Leech lattice. This gives 23-dimensional representations over any field; over fields of characteristic 2 or 3 this can be reduced to a 22-dimensional faithful representation.
The subgroup fixing one of the 759 (= 3·11·23) octads of the Golay code or Steiner system is the octad group 2 4:A 8, order 322560, with orbits of size 8 and 16. The linear group GL(4,2) has an exceptional isomorphism to the alternating group A 8 .
In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup.For example, in the general linear group GL n (n x n invertible matrices), the subgroup of invertible upper triangular matrices is a Borel subgroup.
In 1912 Dehn gave an algorithm that solves both the word and conjugacy problem for the fundamental groups of closed orientable two-dimensional manifolds of genus greater than or equal to 2 (the genus 0 and genus 1 cases being trivial). It is known that the conjugacy problem is undecidable for many classes of groups. Classes of group ...