Subgroup in the context of "Volga Tatars"

In mathematics, specifically in group theory, two groups are commensurable if they differ only by a finite amount, in a precise sense. The commensurator of a subgroup is another subgroup, related to the normalizer.

In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object X is G = Sym(X).

For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space. This article mainly considers symmetry groups in Euclidean geometry, but the concept may also be studied for more general types of geometric structure.

In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses.

In other words, if ${\displaystyle S}$ is a subset of a group ${\displaystyle G}$, then ${\displaystyle \langle S\rangle }$, the subgroup generated by ${\displaystyle S}$, is the smallest subgroup of ${\displaystyle G}$ containing every element of ${\displaystyle S}$, which is equal to the intersection over all subgroups containing the elements of ${\displaystyle S}$; equivalently, ${\displaystyle \langle S\rangle }$ is the subgroup of all elements of ${\displaystyle G}$ that can be expressed as the finite product of elements in ${\displaystyle S}$ and their inverses. (Note that inverses are only needed if the group is infinite; in a finite group, the inverse of an element can be expressed as a power of that element.)

In mathematics, especially in the area of abstract algebra that studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Given a property P, the group G is said to be virtually P if there is a finite index subgroup ${\displaystyle H\leq G}$ such that H has property P.

Common uses for this would be when P is abelian, nilpotent, solvable or free. For example, virtually solvable groups are one of the two alternatives in the Tits alternative, while Gromov's theorem states that the finitely generated groups with polynomial growth are precisely the finitely generated virtually nilpotent groups.

The Awori is a subgroup of the Yoruba people of (heterogeneous origin) speaking a dialect of the Yoruba language.

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

When some object ${\displaystyle X}$ is said to be embedded in another object ${\displaystyle Y}$, the embedding is given by some injective and structure-preserving map ${\displaystyle f:X\rightarrow Y}$. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which ${\displaystyle X}$ and ${\displaystyle Y}$ are instances. In the terminology of category theory, a structure-preserving map is called a morphism.

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.

Gurans (Russian: Гураны) are a Slavo-Mongolic ethnic or subgroup, mainly from Transbaikalia, that formed as a result of mixed marriages between Russians and Buryats (and other indigenous ethnic groups).