Group theoryIn abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.
HalocarbonHalocarbon compounds are chemicals in which one or more carbon atoms are linked by covalent bonds with one or more halogen atoms (fluorine, chlorine, bromine or iodine – group 17) resulting in the formation of organofluorine compounds, organochlorine compounds, organobromine compounds, and organoiodine compounds. Chlorine halocarbons are the most common and are called organochlorides. Many synthetic organic compounds such as plastic polymers, and a few natural ones, contain halogen atoms; they are known as halogenated compounds or organohalogens.
OxohalideIn chemistry, molecular oxohalides (oxyhalides) are a group of chemical compounds in which both oxygen and halogen atoms are attached to another chemical element A in a single molecule. They have the general formula , where X is a halogen. Known oxohalides have fluorine (F), chlorine (Cl), bromine (Br), and/or iodine (I) in their molecules. The element A may be a main group element, a transition element, a rare earth element or an actinide.
GlycosideIn chemistry, a glycoside ˈɡlaɪkəsaɪd is a molecule in which a sugar is bound to another functional group via a glycosidic bond. Glycosides play numerous important roles in living organisms. Many plants store chemicals in the form of inactive glycosides. These can be activated by enzyme hydrolysis, which causes the sugar part to be broken off, making the chemical available for use. Many such plant glycosides are used as medications. Several species of Heliconius butterfly are capable of incorporating these plant compounds as a form of chemical defense against predators.
Trivial groupIn mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually denoted as such: or depending on the context. If the group operation is denoted then it is defined by The similarly defined is also a group since its only element is its own inverse, and is hence the same as the trivial group.
