In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, a group isomorphism is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
between two
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
that sets up a
one-to-one correspondence
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
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
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, 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, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
group homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
w ...
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 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
subgroup
In group theory, a branch of mathematics, given a 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, ''H'' is a subgroup ...
s 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
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
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 number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s under addition,
, is isomorphic to the group of
positive real numbers
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
under multiplication
:
*:
via the isomorphism
.
* The group
of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s (with addition) is a subgroup of
and the
factor group is isomorphic to the group
of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s of
absolute value 1 (under multiplication):
*:
* The
Klein four-group
In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity)
and in which composing any two of the three non-identity elements produces the third one. ...
is isomorphic to the
direct product of two copies of
, and can therefore be written
Another notation is
because it is a
dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...
.
* Generalizing this, for all
odd is isomorphic to the direct product of
and
* If
is an
infinite cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binar ...
, 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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, 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.
* The group
of non-zero complex numbers with multiplication as the 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 learn ...
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 of
equals the order of
If
and
are isomorphic, then
is a
locally finite group In mathematics, in the field of group theory, a locally finite group is a type of group that can be studied in ways analogous to a finite group. Sylow subgroups, Carter subgroups, and abelian subgroups of locally finite groups have been studied. T ...
if and only if
is locally finite.
The number of distinct groups (up to isomorphism) of
order is given by
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
A000001 in the
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 researching 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
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" satisfies 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
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wo ...
is always a conjugacy class (the same or another).
The
composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
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 ''
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
'' 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
In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity)
and in which composing any two of the three non-identity elements produces the third one. ...
. 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
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
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
In finite geometry, the Fano plane (after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and lines ...
, of which the 7 points correspond to the 7 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 automorphism In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a t ...
s.
Non-abelian groups have a non-trivial
inner automorphism
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group it ...
group, and possibly also outer automorphisms.
See also
*
Group isomorphism problem
In abstract algebra, the group isomorphism problem is the decision problem of determining whether two given finite group presentations refer to isomorphic groups.
The isomorphism problem was formulated by Max Dehn, and together with the word pr ...
*
References
*
{{reflist
Group theory
Morphisms