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The roots of periodization come from Hans Selye's model, known as the General adaptation syndrome (GAS). The GAS describes three basic stages of response to stress: (a) the Alarm stage, involving the initial shock of the stimulus on the system, (b) the Resistance stage, involving the adaptation to the stimulus by the system, and (c) the Exhaustion stage, in that repairs are inadequate, and a ...
The additive group of rational numbers (Q, +) is locally cyclic – any pair of rational numbers a/b and c/d is contained in the cyclic subgroup generated by 1/(bd). [2]The additive group of the dyadic rational numbers, the rational numbers of the form a/2 b, is also locally cyclic – any pair of dyadic rational numbers a/2 b and c/2 d is contained in the cyclic subgroup generated by 1/2 max ...
A locally cyclic group is a group in which each finitely generated subgroup is cyclic. An example is the additive group of the rational numbers: every finite set of rational numbers is a set of integer multiples of a single unit fraction, the inverse of their lowest common denominator, and generates as a subgroup a cyclic group of integer ...
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 ...
In the context of group theory, the socle of a group G, denoted soc(G), is the subgroup generated by the minimal normal subgroups of G.It can happen that a group has no minimal non-trivial normal subgroup (that is, every non-trivial normal subgroup properly contains another such subgroup) and in that case the socle is defined to be the subgroup generated by the identity.
Bounded generation is unaffected by passing to a subgroup of finite index: if H is a finite index subgroup of G then G is boundedly generated if and only if H is boundedly generated. Bounded generation goes to extension: if a group G has a normal subgroup N such that both N and G/N are boundedly generated, then so is G itself.
An example of a group that is not complemented (in either sense) is the cyclic group of order p 2, where p is a prime number. This group only has one nontrivial subgroup H, the cyclic group of order p, so there can be no other subgroup L to be the complement of H.
Here, each () / is a cyclic subgroup of Z(p ∞) with p n elements; it contains precisely those elements of Z(p ∞) whose order divides p n and corresponds to the set of p n-th roots of unity. The Prüfer p -groups are the only infinite groups whose subgroups are totally ordered by inclusion.