Bijective group homomorphism
In
abstract algebra, a group isomorphism is a
function between two
groups that sets up a
bijection between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of
group theory, isomorphic groups have the same properties and need not be distinguished.
[1]
Definition and notation
Given two groups and a group isomorphism from to is a
bijective
group homomorphism from to Spelled out, this means that a group isomorphism is a bijective function such that for all and in it holds that
The two groups and are isomorphic if there exists an isomorphism from one to the other.
[1]
[2] This is written
Often shorter and simpler notations can be used. When the relevant group operations are understood, they are omitted and one writes
Sometimes one can even simply write Whether such a notation is possible without confusion or ambiguity depends on context. For example, the equals sign is not very suitable when the groups are both
subgroups of the same group. See also the examples.
Conversely, given a group a set and a
bijection we can make a group by defining
If and then the bijection is an
automorphism (q.v.).
Intuitively, group theorists view two isomorphic groups as follows: For every element of a group there exists an element of such that "behaves in the same way" as (operates with other elements of the group in the same way as ). For instance, if
generates then so does This implies, in particular, that and are in bijective correspondence. Thus, the definition of an isomorphism is quite natural.
An isomorphism of groups may equivalently be defined as an
invertible group homomorphism (the inverse function of a bijective group homomorphism is also a group homomorphism).
Examples
In this section some notable examples of isomorphic groups are listed.
- The group of all
real numbers under addition, , is isomorphic to the group of
positive real numbers under multiplication :
- via the isomorphism .
- 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 (under multiplication):
- The
Klein four-group is isomorphic to the
direct product of two copies of , and can therefore be written Another notation is because it is a
dihedral group.
- Generalizing this, for all
odd is isomorphic to the direct product of and
- If is an
infinite cyclic group, then 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 under addition.
[3]
- The group of non-zero complex numbers with multiplication as the operation is isomorphic to the group mentioned above.
Properties
The
kernel of an isomorphism from to is always {eG}, where eG is the
identity of the group
If and are isomorphic, then is
abelian if and only if is abelian.
If is an isomorphism from to then for any the
order of equals the order of
If and are isomorphic, then is a
locally finite group if and only if is locally finite.
The number of distinct groups (up to isomorphism) of
order is given by
sequence A000001 in the
OEIS. The first few numbers are 0, 1, 1, 1 and 2 meaning that 4 is the lowest order with more than one group.
Cyclic groups
All cyclic groups of a given order are isomorphic to where denotes addition
modulo
Let be a cyclic group and be the order of Letting be a generator of , is then equal to
We will show that
Define
so that
Clearly,
is bijective. Then
which proves that
Consequences
From the definition, it follows that any isomorphism will map the identity element of to the identity element of
that it will map
inverses to inverses,
and more generally,
th powers to
th powers,
and that the inverse map
is also a group isomorphism.
The
relation "being isomorphic" is an
equivalence relation. If is an isomorphism between two groups and then everything that is true about that is only related to the group structure can be translated via into a true ditto statement about and vice versa.
Automorphisms
An isomorphism from a group to itself is called an
automorphism of the group. Thus it is a bijection such that
The
image under an automorphism of a
conjugacy class is always a conjugacy class (the same or another).
The
composition of two automorphisms is again an automorphism, and with this operation the set of all automorphisms of a group denoted by itself forms a group, the
automorphism group of
For all abelian groups there is at least the automorphism that replaces the group elements by their inverses. 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 (which itself is isomorphic to ).
In for a
prime number one non-identity element can be replaced by any other, with corresponding changes in the other elements. The automorphism group is isomorphic to For example, for multiplying all elements of by 3, modulo 7, is an automorphism of order 6 in the automorphism group, because while lower powers do not give 1. Thus this automorphism generates There is one more automorphism with this property: multiplying all elements of by 5, modulo 7. Therefore, these two correspond to the elements 1 and 5 of in that order or conversely.
The automorphism group of is isomorphic to because only each of the two elements 1 and 5 generate so apart from the identity we can only interchange these.
The automorphism group of has order 168, as can be found as follows. 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 For we can choose from 4, which determines the rest. Thus we have automorphisms. They correspond to those of the
Fano plane, of which the 7 points correspond to the 7 non-identity elements. The lines connecting three points correspond to the group operation: and on one line means and See also
general linear group over finite fields.
For abelian groups, all non-trivial automorphisms are
outer automorphisms.
Non-abelian groups have a non-trivial
inner automorphism group, and possibly also outer automorphisms.
See also
References
- Herstein, I. N. (1975). Topics in Algebra (2nd ed.). New York: John Wiley & Sons.
ISBN
0471010901.