Isomorphism of groups

Group isomorphism is a one-to-one correspondence between elements of two groups , preserving group operations. If there is an isomorphism between two groups, the groups are called isomorphic . From the point of view of group theory, isomorphic groups have the same properties and can not be distinguished.

Definition

If two groups ( G , ∗) and ( H , $\circ$ ${\ displaystyle \ circ}$ $\circ$ ) Isomorphism of groups from ( G , ∗) to ( H , $\circ$ ${\ displaystyle \ circ}$ $\circ$ ) Is a bijective homomorphism of groups from G to H.

In other words, an isomorphism of groups is a bijection. $f:G\rightarrow H$ ${\ displaystyle f: G \ rightarrow H}$ $f:G\rightarrow H$ such that for any u and v from G we have

f(u*v)=f(u)\circ f(v)

{\ displaystyle f (u * v) = f (u) \ circ f (v)}

f(u*v)=f(u)\circ f(v)

.

Remarks

Two groups ( G , ∗) and ( H , $\circ$ ${\ displaystyle \ circ}$ $\circ$ ) are isomorphic if there is an isomorphism from one to another. This is written as follows:
$(G,*)\cong (H,\circ )$ ${\ displaystyle (G, *) \ cong (H, \ circ)}$ $(G,*)\cong (H,\circ )$

Shorter and simpler recordings are often used. If group operations do not lead to ambiguity, they are omitted:

G\cong H

{\ displaystyle G \ cong H}

G\cong H

(Sometimes they even write simply G = H. Whether such a record leads to confusion and ambiguity depends on the context. For example, using a sign is equally not very suitable when two groups are subgroups of the same group.)

If H = G and $\circ$ ${\ displaystyle \ circ}$ $\circ$ = ∗, the bijection is an automorphism.

An isomorphism of groups can be defined as a two-sided invertible morphism in the category of groups .

Examples

The group of all real numbers by addition, $(\mathbb {R} ,+)$ ${\ displaystyle (\ mathbb {R}, +)}$ $(\mathbb {R} ,+)$ is isomorphic to the group of all positive real numbers by multiplication $(\mathbb {R} ^{+},\cdot )$ ${\ displaystyle (\ mathbb {R} ^ {+}, \ cdot)}$ $(\mathbb {R} ^{+},\cdot )$ :

(\mathbb {R} ,+)\cong (\mathbb {R} ^{+},\cdot )

{\ displaystyle (\ mathbb {R}, +) \ cong (\ mathbb {R} ^ {+}, \ cdot)}

(\mathbb {R} ,+)\cong (\mathbb {R} ^{+},\cdot )

through isomorphism

f(x)=e^{x}

{\ displaystyle f (x) = e ^ {x}}

(see exhibitor ).

Group $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ integers (by addition) is a subgroup $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ , and the factor group $\mathbb {R} /\mathbb {Z}$ ${\ displaystyle \ mathbb {R} / \ mathbb {Z}}$ isomorphic to the group $S^{1}$ ${\ displaystyle S ^ {1}}$ complex numbers with an absolute value of 1 (by multiplication):

(\mathbb {R} /\mathbb {Z} ,+)\cong (S^{1},\cdot )

{\ displaystyle (\ mathbb {R} / \ mathbb {Z}, +) \ cong (S ^ {1}, \ cdot)}

The isomorphism is given by

f(x+\mathbb {Z} )=e^{2\pi xi}

{\ displaystyle f (x + \ mathbb {Z}) = e ^ {2 \ pi xi}}

for any x from

\mathbb {R}

{\ displaystyle \ mathbb {R}}

.

Klein's fourth group is isomorphic to the direct product of two copies $\mathbb {Z} _{2}=\mathbb {Z} /2\mathbb {Z}$ ${\ displaystyle \ mathbb {Z} _ {2} = \ mathbb {Z} / 2 \ mathbb {Z}}$ (see comparison modulo ), and therefore, can be written as $\mathbb {Z} _{2}\times \mathbb {Z} _{2}$ ${\ displaystyle \ mathbb {Z} _ {2} \ times \ mathbb {Z} _ {2}}$ . Another entry is Dih ₂ because it is a dihedral group .

Generalizing, for all odd n , the group Dih _{2 n} is isomorphic to the direct product Dih _n and Z ₂ .

For some groups, isomorphism can be proved on the basis of the axiom of choice , but such a proof does not show how to construct a concrete isomorphism. Examples:

Group ( $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ , +) is isomorphic to the group ( $\mathbb {C}$ ${\ displaystyle \ mathbb {C}}$ , +) of all complex numbers by addition. ^[one]
Group $(\mathbb {C} ^{*},\cdot )$ ${\ displaystyle (\ mathbb {C} ^ {*}, \ cdot)}$ nonzero multiplication complex numbers is isomorphic to the group S ¹ mentioned above.

Loop groups

If ( G , ∗) is an infinite cyclic group , then ( G , ∗) is isomorphic to integers (by addition). From an algebraic point of view, this means that the set of all integers (by addition) is the only infinite cyclic group.

All finite cyclic groups of a given order are isomorphic $(\mathbb {Z} _{n},+)$ ${\ displaystyle (\ mathbb {Z} _ {n}, +)}$ .

Let G be a cyclic group and n be the order of G. G is the group generated by the element $<x>=\{e,x,...,x^{n-1}\}$ ${\ displaystyle <x> = \ {e, x, ..., x ^ {n-1} \}}$ . We will show that

G\cong (\mathbb {Z} _{n},+)

{\ displaystyle G \ cong (\ mathbb {Z} _ {n}, +)}

Define

\varphi :G\rightarrow \mathbb {Z} _{n}=\{0,1,...,n-1\}

{\ displaystyle \ varphi: G \ rightarrow \ mathbb {Z} _ {n} = \ {0,1, ..., n-1 \}}

, so that

\varphi (x^{a})=a

{\ displaystyle \ varphi (x ^ {a}) = a}

. It's clear that

\varphi

{\ displaystyle \ varphi}

bijectively.

In this way,

\varphi (x^{a}\cdot x^{b})=\varphi (x^{a+b})=a+b=\varphi (x^{a})+\varphi (x^{b})

{\ displaystyle \ varphi (x ^ {a} \ cdot x ^ {b}) = \ varphi (x ^ {a + b}) = a + b = \ varphi (x ^ {a}) + \ varphi (x ^ {b})}

, which proves that

G\cong (\mathbb {Z} _{n},+)

{\ displaystyle G \ cong (\ mathbb {Z} _ {n}, +)}

.

Properties

The kernel of an isomorphism from ( G , ∗) to ( H , $\circ$ ${\ displaystyle \ circ}$ ) is always equal to {e _G }, where e _G is the neutral element of the group ( G , ∗)

If ( G , ∗) is isomorphic to ( H , $\circ$ ${\ displaystyle \ circ}$ ) and if G is Abelian , then H is also Abelian.

If f is an isomorphism between ( G , *) and ( H , $\circ$ ${\ displaystyle \ circ}$ ), then if a belongs to G and has order n , then f (a) has the same order.

If ( G , ∗) is a locally finite group isomorphic to ( H , $\circ$ ${\ displaystyle \ circ}$ ), then ( H , $\circ$ ${\ displaystyle \ circ}$ ) is also locally finite.

Consequences

It follows from the definition that any isomorphism $f:G\rightarrow H$ ${\ displaystyle f: G \ rightarrow H}$ maps the neutral element G to the neutral element H ,

f(e_{G})=e_{H}

{\ displaystyle f (e_ {G}) = e_ {H}}

,

whence the inverse is mapped to the inverse,

f(u^{-1})=\left[f(u)\right]^{-1}

{\ displaystyle f (u ^ {- 1}) = \ left [f (u) \ right] ^ {- 1}}

and the nth degree to the nth degree,

f(u^{n})=\left[f(u)\right]^{n}

{\ displaystyle f (u ^ {n}) = \ left [f (u) \ right] ^ {n}}

for all u from G , and also that the inverse map $f^{-1}:H\rightarrow G$ ${\ displaystyle f ^ {- 1}: H \ rightarrow G}$ is also an isomorphism.

The relation is isomorphic satisfies all axioms of the equivalence relation . If f is an isomorphism of two groups G and H , then all statements that are true for G related to the structure of the group can be transferred by f to the same statements in H , and vice versa.

Automorphisms

An isomorphism from the group ( G , ∗) into itself is called an automorphism of this group. Since isomophism $f:G\rightarrow G$ ${\ displaystyle f: G \ rightarrow G}$ bijective

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

{\ displaystyle f (u) * f (v) = f (u * v)}

.

Automorphism always displays a neutral element in itself. The image of a conjugacy class is always a conjugacy class (the same or different). The image of an element has the same order as the element itself.

The composition of two automorphisms is again an automorphism, and this operation with the set of all automorphisms of the group G , denoted by Aut ( G ), forms a group, a group of automorphisms of G.

For all abelian groups, there is at least an automorphism that takes elements of the group to their inverse. However, in groups where all elements are equal to their inverses, this automorphism is trivial, for example, in the Klein quadruple group (for this group, all permutations of three non-neutral elements of the group are automorphisms, so the isomorphism group is isomorphic to S ₃ and Dih ₃ ) .

In Z _p, for a prime p , one non-neutral element can be replaced by another, with corresponding changes in other elements. The automorphism group is isomorphic to Z _{p - 1} . For example, for n = 7, multiplication of all elements of Z ₇ by 3 (modulo 7) is an automorphism of order 6 in the group of automorphisms, since 3 ⁶ ≡ 1 (modulo 7), and lesser powers of 1 do not. Thus, this automorphism generates Z ₆ . There is another automorphism with this property - the multiplication of all elements of Z ₇ by 5 (modulo 7). Thus, these two automorphisms correspond to elements 1 and 5 of Z ₆ , in this order or vice versa.

The automorphism group Z ₆ is isomorphic to Z ₂ , since only these two elements 1 and 5 generate Z ₆ .

The automorphism group Z ₂ × Z ₂ × Z ₂ = Dih ₂ × Z ₂ is of order 168, which can be shown as follows. All 7 elements that are not neutral play the same role, so we can choose which one plays the role (1,0,0). Any of the remaining six can be selected for the role (0,1,0). These two determine what corresponds to (1,1,0). (0,0,1) we can choose from four, and this choice determines the remaining elements. Thus, we obtain 7 × 6 × 4 = 168 automorphisms. They correspond to automorphisms of the Fano plane , 7 points of which correspond to 7 elements that are not neutral. Lines connecting three points correspond to group operations: a , b , and c on a line mean a + b = c , a + c = b , and b + c = a . See also Complete linear group over a finite field .

For abelian groups, all automorphisms, with the exception of the trivial one, are called .

Non-Abelian groups have nontrivial inner automorphisms , and possibly outer automorphisms.

Notes

↑ Ash. A Consequence of the Axiom of Choice // Journal of the Australian Mathematical Society. - 1973. - T. 19 . - S. 306-308 .

Links

Herstein, IN Topics in Algebra. - 2 edition. - Wiley, 1975 .-- ISBN 0-471-01090-1 ..

[1] Ash. A Consequence of the Axiom of Choice // Journal of the Australian Mathematical Society. - 1973. - T. 19 . - S. 306-308 .