Group isomorphism

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In abstract algebra, a group isomorphism is a function between two groups 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.

Contents 1 Definition and notation 1.1 The converse 2 Examples 3 Consequences 4 Automorphisms

[edit] Definition and notation

Given two groups (G, *) and (H, @), 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 f : G → H such that for all u and v in G it holds that

f (u * v) = f (u) @ f (v).

The two groups (G, *) and (H, @) are isomorphic if an isomorphism exists. This is written:

(G, *) (H, @)

(Here the symbol is displayed with TeX; the Unicode symbol ≅ is not visible with all browsers and browser settings.)

Often shorter and simpler notations can be used. Often there is no ambiguity about the group operation, and it can be omitted:

G 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.

[edit] The converse

Given a group (G, *), a set H, and a bijection f : G → H, we can make H a group (H, @) by defining

f (u) @ f (v) = f (u * v)

If H = G and @ = * then the bijection is an automorphism, see below.

[edit] Examples

The group of all real numbers with addition, (R,+), is isomorphic to the group of all positive real numbers with multiplication (R⁺,×):

(R,+) (R⁺,×)

via the isomorphism

f ( x ) = exp ( x )

(see exponential function).

The group Z of integers (with addition) is a subgroup of R, and the factor group R/Z is isomorphic to the group S¹ of complex numbers of absolute value 1 (with multiplication):

R/Z S¹

An isomorphism is given by

f ( x + Z ) = exp ( 2πxi )

for every x in R.

The Klein four-group is isomorphic to the direct product of two copies of Z₂ = Z / 2Z (see modular arithmetic), and can therefore be written Z₂ × Z₂. Another notation is Dih₂, because it is a dihedral group.

Some groups can be proven to be isomorphic, relying on the axiom of choice, while it is even theoretically impossible to construct concrete isomorphisms. Examples:

• The group ( R , + ) is isomorphic to the group ( C , + ) of all complex numbers with addition.
• The group ( C^* , · ) of non-zero complex numbers with multiplication as operation is isomorphic to the group S¹ mentioned above.

[edit] Consequences

From the definition, it follows that any isomorphism f : G → 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 ^-1 ) = f ( u ) ^-1

for all u in G, and that the inverse map f ^-1 : H → G 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.

[edit] Automorphisms

An isomorphism from a group (G,*) to itself is called an automorphism of this group. Thus it is a bijection f : G → 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 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 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_{p − 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.