# Cyclic group:Z4

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## Contents

## Definition

### Verbal definition

The cyclic group of order 4 is defined as a group with four elements where where the exponent is reduced modulo . In other words, it is the cyclic group whose order is four. It can also be viewed as:

- The quotient group of the group of integers by the subgroup comprising multiples of .
- The multiplicative subgroup of the nonzero complex numbers under multiplication, generated by (a squareroot of ).
- The group of rotational symmetries of the square.

### Multiplication table

This is the multiplication table using multiplicative notation:

Element | (identity element) | (generator) | (generator) | |
---|---|---|---|---|

This is the multiplication table using additive notation, i.e., thinking of the group as the group of integers modulo 4:

Element | (identity element) | (generator) | (generator) | |
---|---|---|---|---|

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 4#Arithmetic functions

## Group properties

Property | Satisfied | Explanation | Comment |
---|---|---|---|

Group of prime power order | Yes | By definition | |

Cyclic group | Yes | By definition | Smallest cyclic group of composite order |

Elementary abelian group | No | Not isomorphic to Klein-four group, which is elementary abelian of order four. | |

Abelian group | Yes | Cyclic implies abelian | |

Nilpotent group | Yes | Abelian implies nilpotent | |

Metacyclic group | Yes | Cyclic implies metacyclic | |

Supersolvable group | Yes | Cyclic implies supersolvable | |

Solvable group | Yes | Abelian implies solvable | |

T-group | Yes | Abelian groups are T-groups | |

Simple group | No | Has normal subgroup of order two | Smallest non-trivial non-simple group. |

Characteristically simple group | No | Has characteristic subgroup of order two | Unique smallest non-trivial non-characteristically simple group. |

## GAP implementation

### Group ID

This finite group has order 4 and has ID 1 among the groups of order 4 in GAP's SmallGroup library. For context, there are groups of order 4. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(4,1)`

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := SmallGroup(4,1);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

`IdGroup(G) = [4,1]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Other descriptions

The group can also be defined using GAP's CyclicGroup function as:

`CyclicGroup(4)`