Cyclic group:Z4: Difference between revisions

Revision as of 14:46, 10 December 2023

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

Verbal definition

The cyclic group of order 4 is defined as a group with four elements $e = x^{0}, x^{1}, x^{2}, x^{3}$ where $x^{l} x^{m} = x^{l + m}$ where the exponent is reduced modulo $4$ . 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 $4$ .
The multiplicative subgroup of the nonzero complex numbers under multiplication, generated by $i$ (a squareroot of $- 1$ ).
The group of rotational symmetries of the square.

Multiplication table

This is the multiplication table using multiplicative notation:

Element	$e$ (identity element)	$x$ (generator)	$x^{2}$	$x^{3}$ (generator)
$e$	$e$	$x$	$x^{2}$	$x^{3}$
$x$	$x$	$x^{2}$	$x^{3}$	$e$
$x^{2}$	$x^{2}$	$x^{3}$	$e$	$x$
$x^{3}$	$x^{3}$	$e$	$x$	$x^{2}$

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

Element	$0$ (identity element)	$1$ (generator)	$2$	$3$ (generator)
$0$	$0$	$1$	$2$	$3$
$1$	$1$	$2$	$3$	$0$
$2$	$2$	$3$	$0$	$1$
$3$	$3$	$0$	$1$	$2$

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

Function	Value	Similar groups	Explanation for function value
underlying prime of p-group	2
order (number of elements, equivalently, cardinality or size of underlying set)	4	groups with same order
prime-base logarithm of order	2	groups with same prime-base logarithm of order
max-length of a group	2		max-length of a group equals prime-base logarithm of order for group of prime power order
chief length	2		chief length equals prime-base logarithm of order for group of prime power order
composition length	2		composition length equals prime-base logarithm of order for group of prime power order
exponent of a group	4	groups with same order and exponent of a group \| groups with same prime-base logarithm of order and exponent of a group \| groups with same exponent of a group
prime-base logarithm of exponent	2	groups with same order and prime-base logarithm of exponent \| groups with same prime-base logarithm of order and prime-base logarithm of exponent \| groups with same prime-base logarithm of exponent
Frattini length	2	groups with same order and Frattini length \| groups with same prime-base logarithm of order and Frattini length \| groups with same Frattini length	Frattini length equals prime-base logarithm of exponent for abelian group of prime power order
minimum size of generating set	1	groups with same order and minimum size of generating set \| groups with same prime-base logarithm of order and minimum size of generating set \| groups with same minimum size of generating set
subgroup rank of a group	1	groups with same order and subgroup rank of a group \| groups with same prime-base logarithm of order and subgroup rank of a group \| groups with same subgroup rank of a group	same as minimum size of generating set since it is an abelian group of prime power order
rank of a p-group	1	groups with same order and rank of a p-group \| groups with same prime-base logarithm of order and rank of a p-group \| groups with same rank of a p-group	same as minimum size of generating set since it is an abelian group of prime power order
normal rank of a p-group	1	groups with same order and normal rank of a p-group \| groups with same prime-base logarithm of order and normal rank of a p-group \| groups with same normal rank of a p-group	same as minimum size of generating set since it is an abelian group of prime power order
characteristic rank of a p-group	1	groups with same order and characteristic rank of a p-group \| groups with same prime-base logarithm of order and characteristic rank of a p-group \| groups with same characteristic rank of a p-group	same as minimum size of generating set since it is an abelian group of prime power order
nilpotency class	1		The group is a nontrivial abelian group
derived length	1		The group is a nontrivial abelian group
Fitting length	1		The group is a nontrivial abelian group -

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 $G$ :

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

Conversely, to check whether a given group $G$ 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)

Internal links

@@ Line 1: / Line 1: @@
 {{particular group}}
-{{group of order|4}}
+[[Category:Cyclic groups]]
-{{smallest|nontrivial non-[[simple group]]}}
+[[importance rank::1| ]]
 ==Definition==
@@ Line 14: / Line 13: @@
 ===Multiplication table===
+This is the multiplication table using multiplicative notation:
 {| class="wikitable" border="1"
@@ Line 23: / Line 24: @@
 |-
 | <math>x^2</math> || <math>x^2</math> || <math>x^3</math> || <math>e</math> || <math>x</math>
+|-
+| <math>x^3</math> || <math>x^3</math> || <math>e</math> || <math>x</math> || <math>x^2</math>
+|}
+This is the multiplication table using additive notation, i.e., thinking of the group as the [[group of integers modulo n|group of integers modulo 4]]:
+{| class="wikitable" border="1"
+! Element !! <math>0</math> ([[identity element]]) !! <math>1</math> (generator) !! <math>2</math> !! <math>3</math> (generator)
+|-
+| <math>0</math> || <math>0</math> || <math>1</math> || <math>2</math> || <math>3</math>
+|-
+| <math>1</math> || <math>1</math> || <math>2</math> || <math>3</math> || <math>0</math>
+|-
+| <math>2</math> || <math>2</math> || <math>3</math> || <math>0</math> || <math>1</math>
+|-
+| <math>3</math> || <math>3</math> || <math>0</math> || <math>1</math> || <math>2</math>
+|}
+==Arithmetic functions==
+{{abelian p-group arithmetic function table|
+underlying prime = 2|
+order = 4|
+order p-log = 2|
+exponent = 4|
+exponent p-log = 2|
+rank = 1}}
+==Group properties==
+{| class="wikitable" border="1"
+!Property !! Satisfied !! Explanation !! Comment
+|-
+|[[Satisfies property::Group of prime power order]] || Yes || By definition ||
+|-
+|[[Satisfies property::Cyclic group]] || Yes || By definition || Smallest cyclic group of composite order
+|-
+|[[Dissatisfies property::Elementary abelian group]] || No || Not isomorphic to [[Klein-four group]], which is elementary abelian of order four. ||
+|-
+|[[Satisfies property::Abelian group]] || Yes || [[Cyclic implies abelian]] ||
+|-
+|[[Satisfies property::Nilpotent group]] || Yes || [[Abelian implies nilpotent]] ||
+|-
+|[[Satisfies property::Metacyclic group]] || Yes || Cyclic implies metacyclic ||
+|-
+|[[Satisfies property::Supersolvable group]] || Yes || Cyclic implies supersolvable ||
+|-
+|[[Satisfies property::Solvable group]] || Yes || [[Abelian implies solvable]] ||
+|-
+|[[Satisfies property::T-group]] || Yes || Abelian groups are T-groups ||
+|-
+|[[Dissatisfies property::Simple group]] || No || Has normal subgroup of order two || Smallest non-trivial non-simple group.
+|-
+|[[Dissatisfies property::Characteristically simple group]] || No || Has characteristic subgroup of order two || Unique smallest non-trivial non-characteristically simple group.
 |}
+==GAP implementation==
+{{GAP ID|4|1}}
+===Other descriptions===
+The group can also be defined using GAP's [[GAP:CyclicGroup|CyclicGroup]] function as:
+<tt>CyclicGroup(4)</tt>
+==Internal links==
+* [[Linear representation theory of cyclic group:Z4]]
+* [[Group cohomology of cyclic group:Z4]]
+* [[Galois extensions for cyclic group:Z4]]