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In
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ' ...
, 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 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 groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
, isomorphic groups have the same properties and need not be distinguished.


Definition and notation

Given two groups (G, *) and (H, \odot), a ''group isomorphism'' from (G, *) to (H, \odot) 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 from G to H. Spelled out, this means that a group isomorphism is a bijective function f : G \to H such that for all u and v in G it holds that f(u * v) = f(u) \odot f(v). The two groups (G, *) and (H, \odot) are isomorphic if there exists an isomorphism from one to the other. This is written (G, *) \cong (H, \odot). Often shorter and simpler notations can be used. When the relevant group operations are understood, they are omitted and one writes G \cong H. 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
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 subgrou ...
s of the same group. See also the examples. Conversely, given a group (G, *), a set H, and a
bijection 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 ...
f : G \to H, we can make H a group (H, \odot) by defining f(u) \odot f(v) = f(u * v). If H = G and \odot = * then the bijection is an
automorphism 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 automorphis ...
(''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. 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, (\R, +), 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 f ...
under multiplication (\R^+, \times): *:(\R, +) \cong (\R^+, \times) via the isomorphism f(x) = e^x. * The group \Z of integers (with addition) is a subgroup of \R, and the factor group \R/\Z is isomorphic to the group S^1 of complex numbers of
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
1 (under multiplication): *:\R/\Z \cong S^1 * 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 In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
of two copies of \Z_2 = \Z/2\Z, and can therefore be written \Z_2 \times \Z_2. Another notation is \operatorname_2, because it is a dihedral group. * Generalizing this, for all odd n, \operatorname_ is isomorphic to the direct product of \operatorname_n and \Z_2. * 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 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 (\R, +) is isomorphic to the group (\Complex, +) of all complex numbers under addition. * The group (\Complex^*, \cdot) of non-zero complex numbers with multiplication as the operation is isomorphic to the group S^1 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 learni ...
of an isomorphism from (G, *) to (H, \odot) is always , where eG is the
identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), an ...
of the group (G, *) If (G, *) and (H, \odot) are isomorphic, then G is
abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...
if and only if H is abelian. If f is an isomorphism from (G, *) to (H, \odot), then for any a \in G, the
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
of a equals the order of f(a). If (G, *) and (H, \odot) are isomorphic, then (G, *) is a locally finite group if and only if (H, \odot) is locally finite. The number of distinct groups (up to isomorphism) of
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
n 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 (\Z_n, +_n), where +_n denotes addition
modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is ...
n. Let G be a cyclic group and n be the order of G. Letting x be a generator of G, G is then equal to \langle x \rangle = \left\. We will show that G \cong (\Z_n, +_n). Define \varphi : G \to \Z_n = \, so that \varphi(x^a) = a. Clearly, \varphi is bijective. Then \varphi(x^a \cdot x^b) = \varphi(x^) = a + b = \varphi(x^a) +_n \varphi(x^b), which proves that G \cong (\Z_n, +_n).


Consequences

From the definition, it follows that any isomorphism f : G \to H will map the identity element of G to the identity element of H, f(e_G) = e_H, that it will map inverses to inverses, f(u^) = f(u)^ \quad \text u \in G, and more generally, nth powers to nth powers, f(u^n)= f(u)^n \quad \text u \in G, and that the inverse map f^ : H \to G is also a group isomorphism. The relation "being isomorphic" satisfies is 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.


Automorphisms

An isomorphism from a group (G, *) to itself is called an
automorphism 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 automorphis ...
of the group. Thus it is a bijection f : G \to G such that f(u) * f(v) = f(u * v). The
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
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 G, denoted by \operatorname(G), 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 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 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
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pr ...
s of the three non-identity elements are automorphisms, so the automorphism group is isomorphic to S_3 (which itself is isomorphic to \operatorname_3). In \Z_p 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 \Z_ For example, for n = 7, multiplying all elements of \Z_7 by 3, modulo 7, is an automorphism of order 6 in the automorphism group, because 3^6 \equiv 1 \pmod 7, while lower powers do not give 1. Thus this automorphism generates \Z_6. There is one more automorphism with this property: multiplying all elements of \Z_7 by 5, modulo 7. Therefore, these two correspond to the elements 1 and 5 of \Z_6, in that order or conversely. The automorphism group of \Z_6 is isomorphic to \Z_2, because only each of the two elements 1 and 5 generate \Z_6, so apart from the identity we can only interchange these. The automorphism group of \Z_2 \oplus \Z_2 \oplus \oplus \Z_2 = \operatorname_2 \oplus \Z_2 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 element corresponds to (1,1,0). For (0,0,1) we can choose from 4, which determines the rest. Thus we have 7 \times 6 \times 4 = 168 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: 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 non-trivial automorphisms are outer automorphisms. 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 itse ...
group, and possibly also outer automorphisms.


See also

* Group isomorphism problem *


References

* {{reflist Group theory Morphisms