Direct Product
If A and B are groups then we can form a new group AxB whose elements are in the cartesian product. AxB = {(a,b) | a in A and b in B}Its operation is component wise.
In mathematics, the direct product is a binary operation that takes two mathematical objects and creates a new one that includes all the information from both original objects. The resulting object is often denoted using an × symbol or the word direct product.
In group theory, the direct product of groups is a way of combining two or more groups to create a new one. Given two groups G and H, their direct product is a new group G × H formed by taking all possible ordered pairs of elements (g, h) where g belongs to G and h belongs to H, and defining a group operation on them in a natural way. Specifically, the operation on (g, h) and (g’, h’) in the direct product G × H is defined as (g, h) * (g’, h’) = (g * g’, h * h’).
In linear algebra, the direct product of matrices is a way of multiplying two or more matrices to create a new one. Given two matrices A and B of dimensions m×n and p×q respectively, their direct product is a new matrix AB formed by taking all possible pairs of entries (aij, bkl) where i, k belong to {1, 2, …, m} and j, l belong to {1, 2, …, q}, and defining a matrix operation on them in a natural way. Specifically, the operation on (aij, bkl) and (a’ij, b’kl) in the direct product AB is defined as (aij, bkl) * (a’ij, b’kl) = (aija’ij, bklb’kl).
The direct product is also used in other branches of mathematics, such as topology and category theory, where it has similar meanings and properties.
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