Finite groups · Family

$\mathrm{C}_{p}$ Cyclic groups of prime order

Exactly the abelian finite simple groups.

Classification coverageFinite simple classification

Definition

The additive group of the prime field Fₚ, equivalently a cyclic group generated by an element of order p.

Parameter domain

Exactly the abelian finite simple groups.

Parameter	Allowed values
p	prime number

Classification sources and conventions

Reference properties

Property	Value	Evidence
Parameter domain	p: prime number	Sources
Scope and restrictions	Exactly the abelian finite simple groups.	Sources
Order	p	Sources

Conventions and use

The trivial group is excluded from the definition of a simple group. [sources]

Names and aliases

Z/pZ · Prime-order cyclic

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Related objects and maps

$\mathrm{C}_{n}$
Finite cyclic groups
Prime-order cyclic subfamily

Claims and sources

Each assertion has a stable identifier within this record and a source locator or derivation. Cite the dataset edition when referring to a corrected or historical claim.

Definition

The additive group of the prime field Fₚ, equivalently a cyclic group generated by an element of order p.

Finite simple groups: Lecture 1 — Introduction and alternating groupsSimple groups, first slideOpen source

family:finite-prime-cyclic#definition

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Parameter domain

p: prime number

Finite simple groups: Lecture 1 — Introduction and alternating groupsSimple groups, first slideOpen source

family:finite-prime-cyclic#parameters

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Scope and restrictions

Exactly the abelian finite simple groups.

Finite simple groups: Lecture 1 — Introduction and alternating groupsSimple groups, first slideOpen source

family:finite-prime-cyclic#restriction

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Order

Finite simple groups: Lecture 1 — Introduction and alternating groupsCFSG statement, first slideOpen source

family:finite-prime-cyclic#order-formula

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Conventions and use

The trivial group is excluded from the definition of a simple group.

Finite simple groups: Lecture 1 — Introduction and alternating groupsSimple groups, first slideOpen source

family:finite-prime-cyclic#conventions-1

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Prime-order cyclic subfamily

family:finite-cyclic

Finite simple groups: Lecture 1 — Introduction and alternating groupsCFSG statement, first slideOpen source

family:finite-prime-cyclic#relation-1

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Cp\mathrm{C}_{p}Cp​Cyclic groups of prime order

Parameter domain

Reference properties

Conventions and use

Names and aliases

Related objects and maps

Claims and sources

$\mathrm{C}_{p}$ Cyclic groups of prime order