In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H × H is a group operation on H. This is often denoted H ≤ G, read as "H is a subgroup of G".
The Volga Tatars or simply Tatars (Tatar: татарлар, romanized: tatarlar; Russian: татары, romanized: tatary) are a Turkic ethnic group native to the Volga-Ural region of western Russia, and contains multiple subgroups. Tatars are the second-largest ethnic group in Russia after ethnic Russians. They are primarily found in Tatarstan, where they make up 53.6% of the population. Their native language is Tatar, and are primarily followers of Sunni Islam.
"Tatar" as an ethnonym has a very long and complicated history, and in the past was often used as an umbrella term for different Turkic and Mongolic tribes. Nowadays it mostly refers exclusively to Volga Tatars (known simply as "Tatars"; Tatarlar), who became its "ultimate bearers" after the founding of Tatar ASSR (1920–1990; now Tatarstan). The ethnogenesis of Volga-Ural Tatars is still debated, but their history is usually connected to the Kipchak-Tatars of Golden Horde (1242–1502), and also to its predecessor, Volga Bulgaria (900s–1200s), whose adoption of Islam is celebrated yearly in Tatarstan. After the collapse of the Golden Horde, ancestors of modern Tatars formed the Khanate of Kazan (1438–1552), which lost its independence to Russia after the Siege of Kazan in 1552.
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 is a subset of a group , then , the subgroup generated by , is the smallest subgroup of containing every element of , which is equal to the intersection over all subgroups containing the elements of ; equivalently, is the subgroup of all elements of that can be expressed as the finite product of elements in 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 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 is said to be embedded in another object , the embedding is given by some injective and structure-preserving map . The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which and 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).
