Group isomorphism
Encyclopedia
In abstract algebra
, a group isomorphism is a function
between two group
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.
), a group isomorphism from (G, *) to (H, 
) is a bijective
group homomorphism
from G to H. Spelled out, this means that a group isomorphism is a bijective function
such that for all u and v in G it holds that
The two groups (G, *) and (H,
) are isomorphic if an isomorphism exists. This is written:
Often shorter and more simple notations can be used. Often there is no ambiguity about the group operation, and it can be omitted:
Sometimes one can even simply write G = H. 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 (G, *), a set H, and a bijection
, we can make H a group (H, 
) by defining
If H = G and
= * then the bijection is an automorphism (q.v.)
Intuitively, group theorists view two isomorphic groups as follows: For every element g of a group G, there exists an element h of H such that h 'behaves in the same way' as g (operates with other elements of the group in the same way as g). For instance, if g generates G, then so does h. This implies in particular that G and H are in bijective correspondence. So the definition of an isomorphism is quite natural.
An isomorphism of groups may equivalently be defined as an invertible morphism
in the category of groups
, where invertible here means has a two-sided inverse.
via the isomorphism
(see exponential function
).
An isomorphism is given by
for every x in
.
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:
.
Let G be a cyclic group and n be the order of G. G is then the group generated by
.
We will show that
Define
Then
will map the identity element of G to the identity element of H,
that it will map inverses to inverses,
and more generally, nth powers to nth powers,
for all u in G,
and that the inverse map
is also a group isomorphism.
The relation "being isomorphic" satisfies all the axioms of an equivalence relation
. If f is an isomorphism between two groups G and H, then everything that is true about G that is only related to the group structure can be translated via f into a true ditto statement about H, and vice versa.
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
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 G, denoted by Aut(G), forms itself a group, the automorphism group of G.
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
. For that group all permutations of the three non-identity elements are automorphisms, so the automorphism group is isomorphic to S3 and Dih3.
In Zp for a prime number p, one non-identity element can be replaced by any other, with corresponding changes in the other elements. The automorphism group is isomorphic to Zp − 1. For example, for n = 7, multiplying all elements of Z7 by 3, modulo 7, is an automorphism of order 6 in the automorphism group, because 36 = 1 ( modulo 7 ), while lower powers do not give 1. Thus this automorphism generates Z6. There is one more automorphism with this property: multiplying all elements of Z7 by 5, modulo 7. Therefore, these two correspond to the elements 1 and 5 of Z6, in that order or conversely.
The automorphism group of Z6 is isomorphic to Z2, because only each of the two elements 1 and 5 generate Z6, so apart from the identity we can only interchange these.
The automorphism group of Z2 × Z2 × Z2 = Dih2 × Z2 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 (1,0,0). Any of the remaining 6 can be chosen to play the role of (0,1,0). This determines which corresponds to (1,1,0). For (0,0,1) we can choose from 4, which determines the rest. Thus we have 7 × 6 × 4 = 168 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: a, b, and c on one line means a+b=c, a+c=b, and b+c=a. See also general linear group over finite fields.
For Abelian groups all automorphisms except the trivial one are called outer automorphisms.
Non-Abelian groups have a non-trivial inner automorphism
group, and possibly also outer automorphisms.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, a group isomorphism is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
between two group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
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 (G, *) and (H,), a group isomorphism from (G, *) to (H, ) is a bijectiveBijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
from G to H. Spelled out, this means that a group isomorphism is a bijective function
such that for all u and v in G it holds that
-
.
The two groups (G, *) and (H,
) are isomorphic if an isomorphism exists. This is written:
Often shorter and more simple notations can be used. Often there is no ambiguity about the group operation, and it can be omitted:
Sometimes one can even simply write G = H. 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 (G, *), a set H, and a bijection
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
, we can make H a group (H, ) by defining
-
.
If H = G and
= * then the bijection is an automorphism (q.v.)Intuitively, group theorists view two isomorphic groups as follows: For every element g of a group G, there exists an element h of H such that h 'behaves in the same way' as g (operates with other elements of the group in the same way as g). For instance, if g generates G, then so does h. This implies in particular that G and H are in bijective correspondence. So the definition of an isomorphism is quite natural.
An isomorphism of groups may equivalently be defined as an invertible morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
in the category of groups
Category of groups
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...
, where invertible here means has a two-sided inverse.
Examples
- The group of all real numberReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s with addition, (
,+), is isomorphic to the group of all positive real numbers with multiplication (
+,×):
via the isomorphism
(see exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
).
- The group
of integerIntegerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s (with addition) is a subgroupSubgroupIn group theory, 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 of G if the restriction of * to H x H is a group operation on H...
of
, and the factor group 
is isomorphic to the group 
of complex numberComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s of absolute valueAbsolute valueIn mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
1 (with multiplication):
An isomorphism is given by
for every x in
.- The Klein four-groupKlein four-groupIn mathematics, the Klein four-group is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2...
is isomorphic to the direct productDirect product of groupsIn the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
of two copies of
(see modular arithmeticModular arithmeticIn mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
), and can therefore be written
. Another notation is Dih2, because it is a dihedral groupDihedral groupIn mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...
.
- Generalizing this, for all odd n, Dih2n is isomorphic with the direct productDirect product of groupsIn the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
of Dihn 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 numberComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s with addition. - The group (
*, ·) of non-zero complex numbers with multiplication as operation is isomorphic to the group S1 mentioned above.
Properties
- The kernelKernel (algebra)In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
of an isomorphism from (G, *) to (H,
), is always {eG} where eG is the identity of the group (G, *)
- If (G, *) is isomorphic to (H,
), and if G is abelianAbelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
then so is H.
- If (G, *) is a group that is isomorphic to (H,
) [where f is the isomorphism], then if a belongs to G and has orderOrder (group theory)In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
n, then so does f(a).
- If (G, *) is a locally finite groupLocally finite groupIn 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....
that is isomorphic to (H,
), then (H, 
) is also locally finite.
- The previous examples illustrate that 'group properties' are always preserved by isomorphisms.
Cyclic groups
All cyclic groups of a given order are isomorphic to.Let G be a cyclic group and n be the order of G. G 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 G to the identity element of H,
that it will map inverses to inverses,
and more generally, nth powers to nth powers,
for all u in G,
and that the inverse map
is also a group isomorphism.The relation "being isomorphic" satisfies all the axioms of an equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
. If f is an isomorphism between two groups G and H, then everything that is true about G that is only related to the group structure can be translated via f into a true ditto statement about H, and vice versa.
Automorphisms
An isomorphism from a group (G,*) to itself is called an automorphismAutomorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
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
Conjugacy class
In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...
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 G, denoted by Aut(G), forms itself a group, the automorphism group of G.
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
Klein four-group
In mathematics, the Klein four-group is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2...
. For that group all permutations of the three non-identity elements are automorphisms, so the automorphism group is isomorphic to S3 and Dih3.
In Zp for a prime number p, one non-identity element can be replaced by any other, with corresponding changes in the other elements. The automorphism group is isomorphic to Zp − 1. For example, for n = 7, multiplying all elements of Z7 by 3, modulo 7, is an automorphism of order 6 in the automorphism group, because 36 = 1 ( modulo 7 ), while lower powers do not give 1. Thus this automorphism generates Z6. There is one more automorphism with this property: multiplying all elements of Z7 by 5, modulo 7. Therefore, these two correspond to the elements 1 and 5 of Z6, in that order or conversely.
The automorphism group of Z6 is isomorphic to Z2, because only each of the two elements 1 and 5 generate Z6, so apart from the identity we can only interchange these.
The automorphism group of Z2 × Z2 × Z2 = Dih2 × Z2 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 (1,0,0). Any of the remaining 6 can be chosen to play the role of (0,1,0). This determines which corresponds to (1,1,0). For (0,0,1) we can choose from 4, which determines the rest. Thus we have 7 × 6 × 4 = 168 automorphisms. They correspond to those of the Fano plane
Fano plane
In finite geometry, the Fano plane is the finite projective plane with the smallest possible number of points and lines: 7 each.-Homogeneous coordinates:...
, of which the 7 points correspond to the 7 non-identity elements. The lines connecting three points correspond to the group operation: a, b, and c on one line means a+b=c, a+c=b, and b+c=a. See also general linear group over finite fields.
For Abelian groups all automorphisms except the trivial one are called outer automorphisms.
Non-Abelian groups have a non-trivial inner automorphism
Inner automorphism
In abstract algebra an inner automorphism is a functionwhich, informally, involves a certain operation being applied, then another one performed, and then the initial operation being reversed...
group, and possibly also outer automorphisms.