Group Isomorphism - Examples

Examples

• The group of all real numbers with addition, (,+), is isomorphic to the group of all positive real numbers with multiplication (+,×):

via the isomorphism

(see exponential function).

• The group of integers (with addition) is a subgroup of, and the factor group is isomorphic to the group of complex numbers of absolute value 1 (with multiplication):

An isomorphism is given by

for every x in .

• The Klein four-group is isomorphic to the direct product of two copies of (see modular arithmetic), and can therefore be written . Another notation is Dih₂, because it is a dihedral group.

• Generalizing this, for all odd n, Dih_2n is isomorphic with the direct product of Dih_n and Z₂.

• If (G, *) is an infinite cyclic group, then (G, *) is isomorphic to the integers (with the addition operation). From an algebraic point of view, this means that the set of all integers (with the addition operation) is the 'only' infinite cyclic group.

Some groups can be proven to be isomorphic, relying on the axiom of choice, but the proof does not indicate how to construct a concrete isomorphism. Examples:

• The group (, +) is isomorphic to the group (, +) of all complex numbers with addition.
• The group (*, ·) of non-zero complex numbers with multiplication as operation is isomorphic to the group S1 mentioned above.