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- 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.
This is the multiplication table using multiplicative notation:
This is the multiplication table using additive notation, i.e., thinking of the group as the group of integers modulo 4:
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 4#Arithmetic functions
|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.|
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:
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:
to have GAP output the group ID, that we can then compare to what we want.
The group can also be defined using GAP's CyclicGroup function as: