In
abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, a group isomorphism is a
function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
between two
group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
s that sets up a one-to-one correspondence 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.
Definition and notation
Given two groups
and
a ''group isomorphism'' from
to
is a
bijective
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

group homomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

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. This is written:
Often shorter and simpler notations can be used. When the relevant group operations are unambiguous 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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 morphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

in the
category of groups
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, where invertible here means has a two-sided inverse.
Examples
In this section some notable examples of isomorphic groups are listed.
* The group of all
real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s with addition,
is isomorphic to the group of
positive real numbers In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
with multiplication
:
*:
via the isomorphism
(see
exponential function
The exponential function is a mathematical function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of ...

).
* The group
of
integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...
s (with addition) is a
subgroup
In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...
of
and the
factor group
A quotient group or factor group is a math
Mathematics (from Greek: ) includes the study of such topics as quantity ( number theory), structure (algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit= ...
is isomorphic to the group
of
complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s of
absolute value
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

1 (with multiplication):
*:
* The
Klein four-group
In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity)
and in which composing any two of the three ...
is isomorphic to the
direct productIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of two copies of
(see
modular arithmetic #REDIRECT Modular arithmetic #REDIRECT Modular arithmetic#REDIRECT Modular arithmetic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure ( ...
), and can therefore be written
Another notation is
because it is a
dihedral group
In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...
.
* Generalizing this, for all odd
is isomorphic with the
direct productIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of
and
* If
is an
infinite cyclic group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...
, 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
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

, but the proof does not indicate how to construct a concrete isomorphism. Examples:
* The group
is isomorphic to the group
of all
complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s with addition.
* The group
of non-zero complex numbers with multiplication as operation is isomorphic to the group
mentioned above.
Properties
The
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...
of an isomorphism from
to
is always where e
G 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
Order, ORDER or Orders may refer to:
* Orderliness
Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...
of
equals the order of
If
and
are isomorphic, then
is
locally finite groupIn mathematics, in the field of group theory, a locally finite group is a type of group (mathematics), group that can be studied in ways analogous to a finite group. Sylow subgroups, Carter subgroups, and abelian subgroups of locally finite groups h ...
if and only if
is locally finite.
The number of distinct groups (up to isomorphism) of order
is given by sequence A000001 in
OEIS
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...

. 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
is then the group generated by
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" satisfies all the axioms of an
equivalence relation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. 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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of this group. Thus it is a bijection
such that
An automorphism always maps the identity to itself. The image under an automorphism of a
conjugacy class
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
is always a conjugacy class (the same or another). The image of an element has the same order as that element.
The composition of two automorphisms is again an automorphism, and with this operation the set of all automorphisms of a group
denoted by
forms itself a group, the 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 inverse this is the trivial automorphism, e.g. in the
Klein four-group
In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity)
and in which composing any two of the three ...
. For that group all permutations of the three non-identity elements are automorphisms, so the automorphism group is isomorphic to
and
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 corresponds to
For
we can choose from 4, which determines the rest. Thus we have
automorphisms. They correspond to those of the
Fano plane
In finite geometry, the Fano plane (after Gino Fano) is the Projective plane#Finite projective planes, finite projective plane of order 2. It is the finite projective plane with the smallest possible number of points and lines: 7 points and 7 li ...

, of which the 7 points correspond to the 7 non-identity elements. The lines connecting three points correspond to the group operation:
on one line means
and
See also
general linear group over finite fields.
For abelian groups all automorphisms except the trivial one are called
outer automorphism In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
s.
Non-abelian groups have a non-trivial
inner automorphism
In abstract algebra an inner automorphism is an automorphism
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...
group, and possibly also outer automorphisms.
See also
*
References
* Herstein, I. N., ''Topics in Algebra'', Wiley; 2 edition (June 20, 1975), .
{{reflist
Group theory
Morphisms