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