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GroupTheory

  

QuasicyclicGroup

  

construct a quasicyclic group for a given prime

 

Calling Sequence

Parameters

Options

Description

Additive Quasicyclic Groups

Multiplicative Quasicyclic Groups

Subgroups of Quasicyclic Groups

Examples

Calling Sequence

QuasicyclicGroup( p )

Parameters

p

-

a prime number

Options

• 

formopt : option of the form form = "multiplicative" or form = "additive" (the default)

Description

• 

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 / is generated. A multiplicative version of the group can be realized by passing the option form = "multiplicative".

Additive Quasicyclic Groups

• 

The default form of a quasicyclic p-group G is an additive group that represents the Sylow p-subgroup of the quotient group / of the additive group of rationals by the integers. Formally, the elements of G are cosets q+ 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 0m 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.

Multiplicative Quasicyclic Groups

• 

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 &ExponentialE;2IkPipn, 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.

Subgroups of Quasicyclic Groups

• 

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.)

Examples

withGroupTheory&colon;

GQuasicyclicGroup5

G5

(1)

Quasicyclic groups are of type QuasicyclicGroup, and also of type QuasicyclicSubgroup.

typeG&comma;&apos;QuasicyclicGroup&apos;

true

(2)

typeG&comma;&apos;QuasicyclicSubgroup&apos;

true

(3)

typeG&comma;&apos;Group&apos;

true

(4)

Not only are quasicyclic groups not finite:

IsFiniteG

false

(5)

They are not even finitely generated.

IsFinitelyGeneratedG

false

(6)

GroupOrderG

(7)

A quasicyclic group is abelian, but not cyclic.

IsAbelianG

true

(8)

IsCyclicG

false

(9)

Elements of an additive quasicyclic p-group are rationals with denominator a power of p.

1225&in;G

true

(10)

ElementOrder1225&comma;G

25

(11)

35&in;G

true

(12)

The group operation is rational addition modulo 1.

1225&plus;35&in;G

true

(13)

CanonicalForm1225&plus;35&comma;G

225

(14)

OperationsG:-`.`1225&comma;35

225

(15)

The rational number 23 is not a member of G because its denominator is not a power of 5.

23&in;G

false

(16)

Every integer belongs to G and represents the group identity.

7&in;G

true

(17)

CanonicalForm7&comma;G

0

(18)

Every element of G has order a power of 5.

IsPGroupG

true

(19)

PGroupPrimeG

5

(20)

Finitely generated subgroups of G are finite and cyclic.

HSubgroup12125&comma;35&comma;G

H5

(21)

typeH&comma;&apos;QuasicyclicGroup&apos;

false

(22)

typeH&comma;&apos;QuasicyclicSubgroup&apos;

true

(23)

IsFiniteH

true

(24)

IsCyclicH

true

(25)

GroupOrderH

125

(26)

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.

DrawSubgroupLatticeH

An additive quasicyclic p-group is isomorphic to the multiplicative quasicyclic p-group (for the same prime p).

MQuasicyclicGroup5&comma;&apos;form&apos;&equals;multiplicative

MC5

(27)

The assign option to the AreIsomorphic command affords you the ability to obtain an explicit isomorphism.

AreIsomorphicG&comma;M&comma;&apos;assign&apos;&equals;&apos;iso&apos;

true

(28)

iso325

&ExponentialE;6I25π

(29)

HSubgroupiso425&comma;M

HC25

(30)

IsSubgroupH&comma;M

true

(31)

IsSubgroupH&comma;G

false

(32)

GroupOrderH

25

(33)

IdentifySmallGroupH

25,1

(34)

Check that H is isomorphic to the cyclic permutation group of the same order.

AreIsomorphicH&comma;CyclicGroup25

true

(35)

LSubgroup425&comma;G

L5

(36)

AreIsomorphicH&comma;L

true

(37)

The elements of H and L are distinct, though they are isomorphic.

ElementsH

1&comma;&ExponentialE;I5π&comma;&ExponentialE;I25π&comma;&ExponentialE;3I5π&comma;&ExponentialE;3I25π&comma;&ExponentialE;7I25π&comma;&ExponentialE;9I25π&comma;&ExponentialE;11I25π&comma;&ExponentialE;13I25π&comma;&ExponentialE;17I25π&comma;&ExponentialE;19I25π&comma;&ExponentialE;21I25π&comma;&ExponentialE;23I25π&comma;&ExponentialE;2I5π&comma;&ExponentialE;2I25π&comma;&ExponentialE;4I5π&comma;&ExponentialE;4I25π&comma;&ExponentialE;6I25π&comma;&ExponentialE;8I25π&comma;&ExponentialE;12I25π&comma;&ExponentialE;14I25π&comma;&ExponentialE;16I25π&comma;&ExponentialE;18I25π&comma;&ExponentialE;22I25π&comma;&ExponentialE;24I25π

(38)

ElementsL

0&comma;15&comma;125&comma;25&comma;225&comma;35&comma;325&comma;45&comma;425&comma;625&comma;725&comma;825&comma;925&comma;1125&comma;1225&comma;1325&comma;1425&comma;1625&comma;1725&comma;1825&comma;1925&comma;2125&comma;2225&comma;2325&comma;2425

(39)

Since quasicyclic groups are infinite, it is not possible to compute all of their elements.

ElementsG

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<denomgthenbreakend ifend do&colon;

On the other hand, iterating over the elements of a finite subgroup of a quasicyclic group terminates.

seqElementOrderx&comma;L&comma;x&equals;L

1&comma;5&comma;25

(40)

You can convert a finite quasicyclic subgroup to a permutation group, a finitely presented group, or to a Cayley table group.

PermutationGroupL

1&comma;3&comma;5&comma;7&comma;9&comma;2&comma;10&comma;11&comma;12&comma;13&comma;4&comma;14&comma;15&comma;16&comma;17&comma;6&comma;18&comma;19&comma;20&comma;21&comma;8&comma;22&comma;23&comma;24&comma;25

(41)

CayleyTableGroupL

< a Cayley table group with 25 elements >

(42)

FPGroupL

_g_g25

(43)

As quasicyclic groups have no maximal subgroups, they are equal to their Frattini subgroups.

FFrattiniSubgroupM

FC5

(44)

IsSubgroupM&comma;FandIsSubgroupF&comma;M

true

(45)

A multiplicative quasicyclic 2-group contains the group generated by the imaginary unit.

TQuasicyclicGroup2&comma;&apos;form&apos;&equals;multiplicative

TC2

(46)

USubgroupI&comma;T

UC4

(47)

IsSubgroupU&comma;T

true

(48)

GroupOrderU

4

(49)

ElementsU

−1&comma;1&comma;−I&comma;I

(50)

Quasicyclic groups for different primes are, of course, non-isomorphic.

AreIsomorphicT&comma;M

false

(51)

AreIsomorphicG&comma;T

false

(52)

See Also

exp

GroupTheory[AreIsomorphic]

GroupTheory[ElementOrder]

GroupTheory[Elements]

GroupTheory[FrattiniSubgroup]

GroupTheory[GroupOrder]

GroupTheory[IdentifySmallGroup]

GroupTheory[IsAbelian]

GroupTheory[IsCyclic]

GroupTheory[IsFinite]

GroupTheory[IsFinitelyGenerated]

GroupTheory[IsPGroup]

GroupTheory[IsSubgroup]

GroupTheory[Operations]

GroupTheory[PGroupPrime]

GroupTheory[QuasicyclicGroup][CanonicalForm]

GroupTheory[Subgroup]

type

with