A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure. For example, the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class as a single entity. It is part of the mathematical field known as group theory.
In abstract algebra, a normal subgroup is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G is normal in G if and only if gng−1 ∈ N for all g ∈ G and n ∈ N. The usual notation for this relation is .
In group theory, a branch of mathematics, 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 usually denoted H ≤ G, read as "H is a subgroup of G".
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group defined over a finite set of symbols consists of the permutations that can be performed on the symbols. Since there are such permutation operations, the order of the symmetric group is .
In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a group that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as a power of g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a generator of the group.
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow (1872) that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups.
The Malayo-Polynesian languages are a subgroup of the Austronesian languages, with approximately 385.5 million speakers. The Malayo-Polynesian languages are spoken by the Austronesian peoples outside of Taiwan, of the island nations of Southeast Asia and the Pacific Ocean, with a smaller number in continental Asia in the areas near the Malay peninsula. Cambodia, Vietnam and the Chinese island Hainan serve as the northwest geographic outlier. Malagasy, spoken in the island of Madagascar off the eastern coast of Africa in the Indian Ocean, is the furthest western outlier. The languages spoken south-westward from central Micronesia until Easter Island are sometimes referred to as the Polynesian languages.
In mathematics, specifically group theory, a nilpotent groupG is a group that has an upper central series that terminates with G. Equivalently, its central series is of finite length or its lower central series terminates with {1}.
In mathematics, specifically group theory, the index of a subgroup H in a group G is the number of left cosets of H in G, or equivalently, the number of right cosets of H in G. The index is denoted or or . Because G is the disjoint union of the left cosets and because each left coset has the same size as H, the index is related to the orders of the two groups by the formula
The Zhuang languages are any of more than a dozen Tai languages spoken by the Zhuang people of Southern China in the province of Guangxi and adjacent parts of Yunnan and Guangdong. The Zhuang languages do not form a monophyletic linguistic unit, as northern and southern Zhuang languages are more closely related to other Tai languages than to each other. Northern Zhuang languages form a dialect continuum with Northern Tai varieties across the provincial border in Guizhou, which are designated as Bouyei, whereas Southern Zhuang languages form another dialect continuum with Central Tai varieties such as Nung, Tay and Caolan in Vietnam. Standard Zhuang is based on the Northern Zhuang dialect of Wuming.
In 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. This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.
Contactin 1, also known as CNTN1, is a protein which in humans is encoded by the CNTN1 gene.
Contactin-2 is a protein that in humans is encoded by the CNTN2 gene.
Neurofascin is a protein that in humans is encoded by the NFASC gene.
Contactin-associated protein-like 2 is a protein that in humans is encoded by the CNTNAP2 gene. Since the most recent reference human genome GRCh38, CNTNAP2 is the longest gene in the human genome
Contactin-4 is a protein that in humans is encoded by the CNTN4 gene.
Contactin-3 is a protein that in humans is encoded by the CNTN3 gene.
Contactin 6 is a protein in humans that is encoded by the CNTN6 gene.
CASPR also known as Contactin associated protein 1, Paranodin and CASPR1 is a protein that in humans is encoded by the CNTNAP1 gene. CASPR is a part of the neurexin family of proteins, hence its another name "Neurexin IV". CASPR is a membrane protein found in the neuronal membrane in the paranodal section of the axon[[]] in myelinated neurons, between the Nodes of Ranvier containing Na+ channels, and juxtaparanode, which contains K+ channels. During myelination, caspr associates with contactin in a cis complex, though its precise role in myelination is not yet understood.
Anti-neurofascin demyelinating diseases refers to health conditions engendered by auto-antibodies against neurofascins, which can produce both central and peripheral demyelination. Some cases of combined central and peripheral demyelination (CCPD) could be produced by them.
