construct a quasicyclic group for a given prime
Additive Quasicyclic Groups
Multiplicative Quasicyclic Groups
Subgroups of Quasicyclic Groups
QuasicyclicGroup( p )
a prime number
formopt : option of the form form = "multiplicative" or form = "additive" (the default)
A quasicyclic group is an infinite abelian group G in which each element has order a power of a single prime number p, and such that every proper subgroup of G is a finite cyclic p-group. They are also called Prüfer p-groups.
Abstractly, a quasicyclic group can be defined as a direct limit of the system of finite cyclic groups of the form Cpk, for positive integers k, in which Cpk is embedded naturally in Cpk+1.
The QuasicyclicGroup( p ) calling sequence constructs a quasicyclic p-group, where p is a prime number. By default, an additive representation that models the Sylow p-subgroup of the quotient group ℚ/ℤ is generated. A multiplicative version of the group can be realized by passing the option form = "multiplicative".
The default form of a quasicyclic p-group G is an additive group that represents the Sylow p-subgroup of the quotient group ℚ/ℤ of the additive group of rationals by the integers. Formally, the elements of G are cosets q+ℤ where q is a rational number whose denominator is a power of the prime p. Maple, however, uses rational representatives as group elements, so that the group operation is ordinary addition of rationals modulo 1. That is, two rationals represent the same group element if their difference is an integer. In particular, any integer is a representative of the identity element of the group.
Each rational of the form kpm, where k and m are integers, has an unique equivalent representative where k and m satisfy 0≤m and k is a non-negative integer with k<pm.
The Operations module for the group implements these conventions. Moreover, the CanonicalForm method of the group returns the unique canonical form of any rational representative of a group element.
An alternative multiplicative form of a quasicyclic p-group is the group of complex p-power roots of unity. These are the complex numbers of the form ⅇ2⁢I⁢k⁢Pipn, where n is a non-negative integer, and k is a positive integer such that k<pn. The group operation is then just ordinary multiplication of complex numbers.
Other more complicated expressions can also represent complex roots of unity. You can use the CanonicalForm method of a quasicyclic group to obtain an expression in the form above, subject to simplifications automatically performed by the exp function.
The Operations module for the group implements these conventions.
Since a proper subgroup of a quasicyclic group is not itself quasicyclic (rather, it is a finite cyclic group of prime power order), in order that its elements remain elements of the parent quasicyclic group, it is represented as a QuasicyclicSubgroup object. The group operations are inherited from the parent quasicyclic group. (In fact, full quasicyclic groups are also represented as QuasicyclicSubgroup objects.)
G ≔ QuasicyclicGroup⁡5
Quasicyclic groups are of type QuasicyclicGroup, and also of type QuasicyclicSubgroup.
Not only are quasicyclic groups not finite:
They are not even finitely generated.
A quasicyclic group is abelian, but not cyclic.
Elements of an additive quasicyclic p-group are rationals with denominator a power of p.
The group operation is rational addition modulo 1.
The rational number 23 is not a member of G because its denominator is not a power of 5.
Every integer belongs to G and represents the group identity.
Every element of G has order a power of 5.
Finitely generated subgroups of G are finite and cyclic.
H ≔ Subgroup⁡12125,35,G
The subgroup lattice of a quasicyclic group is a chain, infinite in length. However, we can visualize the subgroup lattice of finite subgroups of quasicyclic groups.
An additive quasicyclic p-group is isomorphic to the multiplicative quasicyclic p-group (for the same prime p).
M ≔ QuasicyclicGroup⁡5,'form'=multiplicative
The assign option to the AreIsomorphic command affords you the ability to obtain an explicit isomorphism.
H ≔ Subgroup⁡iso⁡425,M
Check that H is isomorphic to the cyclic permutation group of the same order.
L ≔ Subgroup⁡425,G
The elements of H and L are distinct, though they are isomorphic.
Since quasicyclic groups are infinite, it is not possible to compute all of their elements.
Error, (in GroupTheory:-Generators) group is not finitely generated
Similarly, you can iterate over the elements of a quasicyclic group but, as the group is infinite, you need to provide a termination condition, as illustrated in the following example.
forginGdoif1000<denom⁡gthenbreakend ifend do:
On the other hand, iterating over the elements of a finite subgroup of a quasicyclic group terminates.
You can convert a finite quasicyclic subgroup to a permutation group, a finitely presented group, or to a Cayley table group.
< a Cayley table group with 25 elements >
As quasicyclic groups have no maximal subgroups, they are equal to their Frattini subgroups.
F ≔ FrattiniSubgroup⁡M
A multiplicative quasicyclic 2-group contains the group generated by the imaginary unit.
T ≔ QuasicyclicGroup⁡2,'form'=multiplicative
U ≔ Subgroup⁡I,T
Quasicyclic groups for different primes are, of course, non-isomorphic.
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