Cyclic group:Z4

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This particular group is a finite group of order: 4 This particular group is the smallest (in terms of order): nontrivial non-simple group

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)
$e$	$e$	$x$	$x^{2}$	$x^{3}$
$x$	$x$	$x^{2}$	$x^{3}$	$e$
$x^{2}$	$x^{2}$	$x^{3}$	$e$	$x$

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)
$0$	$0$	$1$	$2$	$3$
$1$	$1$	$2$	$3$	$0$
$2$	$2$	$3$	$0$	$1$

Arithmetic functions

Function	Value	Explanation
order	4
exponent	4
nilpotency class	1	The group is abelian.
derived length	1	The group is abelian.
Frattini length	2	The Frattini subgroup has order two.
Fitting length	1	The group is abelian, hence nilpotent.
minimum size of generating set	1
subgroup rank	1	Cyclic, so every subgroup is cyclic.
max-length	2
rank as p-group	1
normal rank	1
characteristic rank of a p-group	1

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.

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