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  2. Group theory - Wikipedia

    en.wikipedia.org/wiki/Group_theory

    Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry.The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields.

  3. Direct product of groups - Wikipedia

    en.wikipedia.org/wiki/Direct_product_of_groups

    If E denotes the trivial group, G ≅ G × E ≅ E × G for any groups G. The order of a direct product G × H is the product of the orders of G and H: | G × H | = | G | | H |. This follows from the formula for the cardinality of the cartesian product of sets. The order of each element (g, h) is the least common multiple of the orders of g and ...

  4. Group (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Group_(mathematics)

    The set is called the underlying set of the group, and the operation is called the group operation or the group law. A group and its underlying set are thus two different mathematical objects. To avoid cumbersome notation, it is common to abuse notation by using the same symbol to denote both. This reflects also an informal way of thinking ...

  5. Centralizer and normalizer - Wikipedia

    en.wikipedia.org/wiki/Centralizer_and_normalizer

    E.g. the Weyl group of a compact Lie group G with a torus T is defined as W(G,T) = N G (T)/C G (T), and especially if the torus is maximal (i.e. C G (T) = T) it is a central tool in the theory of Lie groups. C G (C G (S)) contains S, but C G (S) need not contain S. Containment occurs exactly when S is abelian. If H is a subgroup of G, then N G ...

  6. Mathematics of Sudoku - Wikipedia

    en.wikipedia.org/wiki/Mathematics_of_Sudoku

    One also says that the latter is a quotient group of the former, because some once different elements become equal in the new group. However, it is also a subgroup, because we can simply fill the missing component with 0 {\displaystyle 0} to get back to Z n ⊕ Z n {\displaystyle \mathbb {Z} _{n}\oplus \mathbb {Z} _{n}} .

  7. Abstract algebra - Wikipedia

    en.wikipedia.org/wiki/Abstract_algebra

    The permutations of the Rubik's Cube form a group, a fundamental concept within abstract algebra.. In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements. [1]

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  9. Algebraic group - Wikipedia

    en.wikipedia.org/wiki/Algebraic_group

    The special linear group is an algebraic group: it is given by the algebraic equation () = in the affine space (identified with the space of -by-matrices), multiplication of matrices is regular and the formula for the inverse in terms of the adjugate matrix shows that inversion is regular as well on matrices with determinant 1.