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specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as...

factorization of elements of the automorphism group of a free group Normal form for free groups and free product of groups Free product von Dyck, Walther...

combinatorial group theory, a normal form for a free group over a set of generators or for a free product of groups is a representation of an element by...

free abelian groups are used to define chain groups, and in algebraic geometry they are used to define divisors. The elements of a free abelian group...

mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H...

semidirect product: an inner semidirect product is a particular way in which a group can be made up of two subgroups, one of which is a normal subgroup...

isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R. As a simple example, the cyclic group of order n has the presentation...

direct product. This subset does indeed form a group, and for a finite set of groups {Hi} the external direct sum is equal to the direct product. If G...

about the finiteness of groups in a particular family. The free Burnside group of rank m and exponent n, denoted B(m, n), is a group with m distinguished...

theory of covering groups and locally isomorphic groups. A discrete normal subgroup of a connected group G necessarily lies in the center of G and is therefore...

smaller groups, namely a nontrivial normal subgroup and the corresponding quotient group. This process can be repeated, and for finite groups one eventually...

is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian...

order. Subgroups of symmetric groups are called permutation groups and are widely studied because of their importance in understanding group actions, homogeneous...

result can also be stated as "any subgroup of index 2 is normal", and in this form it applies also to infinite groups. Furthermore, if p {\displaystyle p} is...

peroxide, and hydroxyl radical, commonly associated with cell damage. ROS form as a natural by-product of the normal metabolism of oxygen and have important...

semidirect product and direct product of the cyclic groups. In the solvable group, C 4 {\displaystyle \mathbb {C} _{4}} is not a normal subgroup. A group G is...

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...

In mathematics, many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often...

In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time...

In group theory, the wreath product is a special combination of two groups based on the semidirect product. It is formed by the action of one group on...