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In abstract algebra, a cyclic group or monogenous group is a group, denoted Cn (also frequently Z {\displaystyle \mathbb {Z} } n or Zn, not to be confused...

finitely generated group is virtually nilpotent if and only if it has polynomial growth. Any free group. Any virtually cyclic group. Any semidirect product...

of a cyclic group by a cyclic group. Examples of polycyclic groups include finitely generated abelian groups, finitely generated nilpotent groups, and...

classes of groups: cyclic < abelian < nilpotent < supersolvable < polycyclic < solvable < finitely generated group. A group G is called virtually solvable...

of virtually cyclic subgroups of G. So assuming the Farrell–Jones conjecture is true, it is possible to restrict computations to virtually cyclic subgroups...

known ketose is fructose; it mostly exists as a cyclic hemiketal, which masks the ketone functional group. Fatty acid synthesis proceeds via ketones. Acetoacetate...

group is finite). For n ≥ 2 {\displaystyle n\geq 2} the group Out(Fn) is acylindrically hyperbolic. By a result of Osin, every non virtually cyclic group...

graph is 0 {\displaystyle 0} -hyperbolic. It follows that any group which is virtually cyclic (contains a copy of Z {\displaystyle \mathbb {Z} } of finite...

of the Heisenberg group by a central cyclic subgroup. Discrete subgroups of classical Lie groups (for example lattices or thin groups) are also examples...

introduce alkyl or aryl groups onto carbonyl compounds. These compounds can then undergo further reactions leading to the formation of cyclic structures, including...

space and r,s are slopes on T such that their Dehn fillings have cyclic fundamental group, then the distance between r and s (the minimal number of times...

afford linear or cyclic products. Linear products are terminated with silanol groups: n R2Si(OH)2 → H(R2SiO)nOH + (n − 1) H2O Cyclic products have no...

fundamental group π1(M): If π1(M) is finite then the geometric structure on M is spherical, and M is compact. If π1(M) is virtually cyclic but not finite...

proved in 1967 the following complementary fact: A group G {\displaystyle G} is virtually infinite cyclic if and only if it has a finite normal subgroup W...

symmetric groups, which are themselves examples of wreath products. (There is no factor for re-arrangements of the center faces, because on virtually all Rubik's...

p-Methoxyphenyl (PMP) group – Removed by ammonium cerium(IV) nitrate (CAN) The most common protecting groups for carbonyls are acetals and typically cyclic acetals...

Sapir, Mark V. (2002), "Non-amenable finitely presented torsion-by-cyclic groups", Publ. Math. Inst. Hautes Études Sci., 96: 43–169, arXiv:math/0208237...

in rank-one p-adic groups are virtually free groups they are very non-rigid. Let T {\displaystyle T} be a tree with a cocompact group of automorphisms;...

polynomials over a ring R is the group ring of the infinite cyclic group Z over R. 3. Let Q be the quaternion group with elements { e , e ¯ , i , i ¯...

is, not virtually cyclic) virtually free group which is co-Hopfian. Thus a subgroup of finite index in a finitely generated co-Hopfian group need not...