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Let be a *-algebra. Then: An element is normal if and only if the *-subalgebra generated by , meaning the smallest *-algebra containing , is commutative. [2]Every element can be uniquely decomposed into a real and imaginary part, which means there exist self-adjoint elements ,, such that = +, where denotes the imaginary unit.
A Fourier series (/ ˈ f ʊr i eɪ,-i ər / [1]) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. [2] By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are ...
In the quaternion group of order 8, each of the cyclic subgroups of order 4 is normal, but none of these are characteristic. However, the subgroup, {1, −1}, is characteristic, since it is the only subgroup of order 2. If n > 2 is even, the dihedral group of order 2n has 3 subgroups of index 2, all of which are normal. One of these is the ...
Grove, Larry C. (2012), "Theorem 1.11 (The Fundamental Homomorphism Theorem)", Algebra, Dover Books on Mathematics, Courier Corporation, p. 11, ISBN 9780486142135 Jacobson, Nathan (2012), "Fundamental theorem on homomorphisms of Ω-algebras", Basic Algebra II , Dover Books on Mathematics (2nd ed.), Courier Corporation, p. 62, ISBN 9780486135212
In Boolean algebra, the algebraic normal form (ANF), ring sum normal form (RSNF or RNF), ...
The lattice Con(A) of all congruence relations on an algebra A is algebraic. John M. Howie described how semigroup theory illustrates congruence relations in universal algebra: In a group a congruence is determined if we know a single congruence class, in particular if we know the normal subgroup which is the class containing the identity.