is a bijective group homomorphism from
Spelled out, this means that a group isomorphism is a bijective function
Often shorter and simpler notations can be used.
When the relevant group operations are understood, they are omitted and one writes
Whether such a notation is possible without confusion or ambiguity depends on context.
(operates with other elements of the group in the same way as
Thus, the definition of an isomorphism is quite natural.
In this section some notable examples of isomorphic groups are listed.
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 kernel of an isomorphism from
is a locally finite group if and only if
The number of distinct groups (up to isomorphism) of order
The first few numbers are 0, 1, 1, 1 and 2 meaning that 4 is the lowest order with more than one group.
All cyclic groups of a given order are isomorphic to
denotes addition modulo
that is only related to the group structure can be translated via
into a true ditto statement about
to itself is called an automorphism of the group.
However, in groups where all elements are equal to their inverses this is the trivial automorphism, e.g. in the Klein four-group.
For that group all permutations of the three non-identity elements are automorphisms, so the automorphism group is isomorphic to
The automorphism group is isomorphic to
while lower powers do not give 1.
There is one more automorphism with this property: multiplying all elements of
All 7 non-identity elements play the same role, so we can choose which plays the role of
Any of the remaining 6 can be chosen to play the role of (0,1,0).
This determines which element corresponds to
we can choose from 4, which determines the rest.
The lines connecting three points correspond to the group operation:
See also general linear group over finite fields.
Non-abelian groups have a non-trivial inner automorphism group, and possibly also outer automorphisms.