Groupprops, The Group Properties Wiki (pre-alpha)

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From Groupprops

Definition

Let p be a prime number. The p-quasicyclic group is defined in the following equivalent ways:

• It is the group, under multiplication, of all complex (pⁿ)^th roots of unity for all n.
• It is the quotient where L is the group of all rational numbers that can be expressed with denominator a power of p.
• It is the direct limit of the chain of groups:

.

where the maps are multiplication by p maps.

The quasicyclic group is Abelian, locally finite, and locally cyclic, any two subgroups of it are comparable, and every subgroup is characteristic.