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Brauer's third main theorem (Brauer 1964, theorem3) states that when Q is a p-subgroup of the finite group G, and H is a subgroup of G containing () and contained in (), then the principal block of H is the only Brauer correspondent of the principal block of G (where the blocks referred to are calculated in characteristic p).
Equivalence class: given an equivalence relation, [] often denotes the equivalence class of the element x. 3. Integral part : if x is a real number , [ x ] {\displaystyle [x]} often denotes the integral part or truncation of x , that is, the integer obtained by removing all digits after the decimal mark .
The upper p-series is formed by taking π 2i−1 = p′ and π 2i = p; there is also a lower p-series. A finite group is said to be p-nilpotent if and only if it is equal to its own p′,p-core. A finite group is said to be p-soluble if and only if it is equal to some term of its upper p-series; its p-length is the length of its upper p-series.
A partially balanced incomplete block design with n associate classes (PBIBD(n)) is a block design based on a v-set X with b blocks each of size k and with each element appearing in r blocks, such that there is an association scheme with n classes defined on X where, if elements x and y are ith associates, 1 ≤ i ≤ n, then they are together ...
The Schur complement arises when performing a block Gaussian elimination on the matrix M.In order to eliminate the elements below the block diagonal, one multiplies the matrix M by a block lower triangular matrix on the right as follows: = [] [] [] = [], where I p denotes a p×p identity matrix.
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The p orbital can hold a maximum of six electrons, hence there are six columns in the p-block. Elements in column 13, the first column of the p-block, have one p-orbital electron. Elements in column 14, the second column of the p-block, have two p-orbital electrons. The trend continues this way until column 18, which has six p-orbital electrons.
Given an eigenvalue λ, every corresponding Jordan block gives rise to a Jordan chain of linearly independent vectors p i, i = 1, ..., b, where b is the size of the Jordan block. The generator, or lead vector, p b of the chain is a generalized eigenvector such that (A − λI) b p b = 0. The vector p 1 = (A − λI) b−1 p b is an ordinary ...