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A cyclic group is a group which is equal to one of its cyclic subgroups: G = g for some element g, called a generator of G. For a finite cyclic group G of order n we have G = {e, g, g 2, ... , g n−1}, where e is the identity element and g i = g j whenever i ≡ j (mod n); in particular g n = g 0 = e, and g −1 = g n−1.
The lattice of subgroups of the infinite cyclic group can be described in the same way, as the dual of the divisibility lattice of all positive integers. If the infinite cyclic group is represented as the additive group on the integers, then the subgroup generated by d is a subgroup of the subgroup generated by e if and only if e is a divisor ...
Cycle, a set equipped with a cyclic order. Necklace (combinatorics), an equivalence classes of cyclically ordered sequences of symbols modulo certain symmetries; Cyclic (mathematics), a list of mathematics articles with "cyclic" in the title; Cyclic group, a group generated by a single element
Cyclic succession is a pattern of vegetation change in which in a small number of species tend to replace each other over time in the absence of large-scale disturbance. Observations of cyclic replacement have provided evidence against traditional Clementsian views of an end-state climax community with stable species compositions .
It can be shown that a finite p-group with this property (every abelian subgroup is cyclic) is either cyclic or a generalized quaternion group as defined above. [12] Another characterization is that a finite p-group in which there is a unique subgroup of order p is either cyclic or a 2-group isomorphic to generalized quaternion group. [13]
The 15-crown-5 crown ether, a cyclic oligomer, and its monomer, ethylene oxide.. In chemistry and biochemistry, an oligomer (/ ə ˈ l ɪ ɡ ə m ər / ⓘ) is a molecule that consists of a few repeating units which could be derived, actually or conceptually, from smaller molecules, monomers.
A cyclic nucleotide (cNMP) is a single-phosphate nucleotide with a cyclic bond arrangement between the sugar and phosphate groups. Like other nucleotides, cyclic nucleotides are composed of three functional groups: a sugar, a nitrogenous base , and a single phosphate group.
Cyclic photophosphorylation occurs in both aerobic and anaerobic conditions, driven by the main primary source of energy available to living organisms, which is sunlight. All organisms produce a phosphate compound, ATP , which is the universal energy currency of life.