TheInfoList

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 $\left(G, *\right)$ and $\left(H, \odot\right),$ a ''group isomorphism'' from $\left(G, *\right)$ to $\left(H, \odot\right)$ 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 $G$ 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 $\left(G, *\right)$ and $\left(H, \odot\right)$ 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 unambiguous 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 subgroups of the same group. See also the examples. Conversely, given a group $\left(G, *\right),$ a set $H,$ 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 ...

$f : G \to H,$ we can make $H$ a group $\left(H, \odot\right)$ by defining $f(u) \odot f(v) = f(u * v).$ If $H$ = $G$ and $\odot = *$ 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. 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, $\left(\R, +\right),$ 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 $\left\left(\R^+, \times\right\right)$: *:$\left(\R, +\right) \cong \left\left(\R^+, \times\right\right)$ via the isomorphism $f\left(x\right) = e^x$ (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 $\Z$ 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 ...
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 $\R,$ 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= ...
$\R/\Z$ is isomorphic to the group $S^1$ 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): *:$\R/\Z \cong S^1$ * 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 $\Z_2 = \Z/2\Z$ (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 $\Z_2 \times \Z_2.$ Another notation is $\operatorname_2,$ 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 $n,$ $\operatorname_$ 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 $\operatorname_n$ and $Z_2.$ * If $\left(G, *\right)$ 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 $\left(G, *\right)$ 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 $\left(\R, +\right)$ is isomorphic to the group $\left(\Complex, +\right)$ 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 $\left\left(\Complex^*, \cdot\right\right)$ of non-zero complex numbers with multiplication as 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 learnin ...
of an isomorphism from $\left(G, *\right)$ to $\left(H, \odot\right),$ is always where eG is the identity of the group $\left(G, *\right)$ If $\left(G, *\right)$ and $\left(H, \odot\right)$ are isomorphic, then $G$ is abelian if and only if $H$ is abelian. If $f$ is an isomorphism from $\left(G, *\right)$ to $\left(H, \odot\right),$ then for any $a \in G,$ 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 $a$ equals the order of $f\left(a\right).$ If $\left(G, *\right)$ and $\left(H, \odot\right)$ are isomorphic, then $\left(G, *\right)$ 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 $\left(H, \odot\right)$ is locally finite. The number of distinct groups (up to isomorphism) of order $n$ 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 $\left(\Z_n, +_n\right),$ where $+_n$ denotes addition modulo $n.$ Let $G$ be a cyclic group and $n$ be the order of $G.$ $G$ is then the group generated by $\langle x \rangle = \left\.$ We will show that $G \cong \left(\Z_n, +_n\right).$ Define $\varphi : G \to \Z_n = \,$ so that $\varphi\left\left(x^a\right\right) = a.$ Clearly, $\varphi$ is bijective. Then $\varphi\left(x^a \cdot x^b\right) = \varphi\left(x^\right) = a + b = \varphi\left(x^a\right) +_n \varphi\left(x^b\right),$ which proves that $G \cong \left\left(\Z_n, +_n\right\right).$

# 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 $H,$ $f\left(e_G\right) = e_H$ that it will map inverses to inverses, and more generally, $b$th powers to $n$th powers, and that the inverse map $f^ : H \to G$ 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 $f$ 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 $\left(G, *\right)$ 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 $f : G \to G$ such that $f(u) * f(v) = f(u * v).$ 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 $G,$ denoted by $\operatorname\left(G\right),$ forms itself a group, the 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 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 $S_3$ and $\operatorname_3.$ 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 $\left(1,0,0\right).$ Any of the remaining 6 can be chosen to play the role of (0,1,0). This determines which corresponds to $\left(1,1,0\right).$ For $\left(0,0,1\right)$ 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 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: $a, b, \text c$ 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 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.