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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 :
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:
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.
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.
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.
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.
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.
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 ...