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  2. Word problem for groups - Wikipedia

    en.wikipedia.org/wiki/Word_problem_for_groups

    This gives the following criterion for the uniform solvability of the word problem for a class of finitely presented groups: To solve the uniform word problem for a class of groups, it is sufficient to find a recursive function ⁠ (,) ⁠ that takes a finite presentation for a group and a word ⁠ ⁠ in the generators of , such that whenever :

  3. Solvable group - Wikipedia

    en.wikipedia.org/wiki/Solvable_group

    In fact, all supersolvable groups are finitely generated, and an abelian group is supersolvable if and only if it is finitely generated. The alternating group A 4 is an example of a finite solvable group that is not supersolvable. If we restrict ourselves to finitely generated groups, we can consider the following arrangement of classes of groups:

  4. Group theory - Wikipedia

    en.wikipedia.org/wiki/Group_theory

    Finite groups can be described by writing down the group table consisting of all possible multiplications g • h. A more compact way of defining a group is by generators and relations, also called the presentation of a group. Given any set F of generators {}, the free group generated by F surjects onto the group G.

  5. Group (mathematics) - Wikipedia

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

    In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a general group. Lie groups appear in symmetry groups in geometry, and also in the Standard Model of particle physics.

  6. Presentation of a group - Wikipedia

    en.wikipedia.org/wiki/Presentation_of_a_group

    To see this, given a group G, consider the free group F G on G. By the universal property of free groups, there exists a unique group homomorphism φ : F G → G whose restriction to G is the identity map. Let K be the kernel of this homomorphism. Then K is normal in F G, therefore is equal to its normal closure, so G | K = F G /K.

  7. Representation theory of finite groups - Wikipedia

    en.wikipedia.org/wiki/Representation_theory_of...

    Nevertheless, groups acting on other groups or on sets are also considered. For more details, please refer to the section on permutation representations. Other than a few marked exceptions, only finite groups will be considered in this article. We will also restrict ourselves to vector spaces over fields of characteristic zero.

  8. Direct product of groups - Wikipedia

    en.wikipedia.org/wiki/Direct_product_of_groups

    The resulting algebraic object satisfies the axioms for a group. Specifically: Associativity The binary operation on G × H is associative. Identity The direct product has an identity element, namely (1 G, 1 H), where 1 G is the identity element of G and 1 H is the identity element of H.

  9. Burnside's theorem - Wikipedia

    en.wikipedia.org/wiki/Burnside's_theorem

    Let p a q b be the smallest product of two prime powers, such that there is a non-solvable group G whose order is equal to this number. G is a simple group with trivial center and a is not zero. If G had a nontrivial proper normal subgroup H , then (because of the minimality of G ), H and G / H would be solvable, so G as well, which would ...