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.
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Definition
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).
Examples
The group of all real numbers with addition, (R,+), is isomorphic to the group of all positive real numbers with multiplication (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); 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 Z/2Z (see modular arithmetic).
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. For these last two examples, one cannot construct concrete isomorphisms; the proofs rely on the axiom of choice.
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(eG) = eH
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 G and H, then everything that is true about G can be translated via f into a true statement about H, and vice versa.
Automorphisms
An isomorphism from a group G to G itself is called an automorphism of G. The composition of two automorphism 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.