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The character table does not in general determine the group up to isomorphism: for example, the quaternion group and the dihedral group of order 8 have the same character table. Brauer asked whether the character table, together with the knowledge of how the powers of elements of its conjugacy classes are distributed, determines a finite group ...
The finite group notation used is: Z n: cyclic group of order n, D n: dihedral group isomorphic to the symmetry group of an n–sided regular polygon, S n: symmetric group on n letters, and A n: alternating group on n letters. The character tables then follow for all groups. The rows of the character tables correspond to the irreducible ...
The space of complex-valued class functions of a finite group G has a natural inner product: , := | | () ¯ where () ¯ denotes the complex conjugate of the value of on g.With respect to this inner product, the irreducible characters form an orthonormal basis for the space of class functions, and this yields the orthogonality relation for the rows of the character table:
The composition factors of the projective indecomposable modules may be calculated as follows: Given the ordinary irreducible and irreducible Brauer characters of a particular finite group, the irreducible ordinary characters may be decomposed as non-negative integer combinations of the irreducible Brauer characters. The integers involved can ...
The character table does not in general determine the group up to isomorphism: for example, the quaternion group Q and the dihedral group of 8 elements, D 4, have the same character table. Brauer asked whether the character table, together with the knowledge of how the powers of elements of its conjugacy classes are distributed, determines a ...
The conclusion from applying character theory to the group G is that G has the following structure: there are primes p>q such that (p q –1)/(p–1) is coprime to p–1 and G has a subgroup given by the semidirect product PU where P is the additive group of a finite field of order p q and U its elements of norm 1.
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An example of a G-structure is an almost complex structure, that is, a reduction of a structure group of an even-dimensional manifold to GL(n,C). Such a reduction is uniquely determined by a C ∞-linear endomorphism J ∈ End(TM) such that J 2 = −1. In this situation, the torsion can be computed explicitly as follows.