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Group codes consist of linear block codes which are subgroups of , where is a finite Abelian group. A systematic group code C {\displaystyle C} is a code over G n {\displaystyle G^{n}} of order | G | k {\displaystyle \left|G\right|^{k}} defined by n − k {\displaystyle n-k} homomorphisms which determine the parity check bits.
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 ...
As well as having applications to group theory, modular representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory [citation needed], combinatorics and number theory. Within finite group theory, character-theoretic results proved by Richard Brauer using modular representation theory played an ...
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 ...
Furthermore, a class function on is a character of if and only if it can be written as a linear combination of the distinct irreducible characters with non-negative integer coefficients: if is a class function on such that = + + where non-negative integers, then is the character of the direct sum of the representations corresponding to .
The number of these irreducibles is equal to the number of conjugacy classes of G. The above fact can be explained by character theory. Recall that the character of the regular representation χ(g) is the number of fixed points of g acting on the regular representation V. It means the number of fixed points χ(g) is zero when g is not id and |G ...
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The order of an element of a group (also called period length or period) is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication, the order of an element a of a group, is thus the smallest positive integer m such that a m = e, where e denotes the identity element of the group, and a m denotes ...