Isomorphism

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In abstract algebra, an isomorphism (Greek: ἴσος isos "equal", and μορφή morphe "shape") is a bijective map f such that both f and its inverse f −1 are homomorphisms, i.e., structure-preserving mappings. In the more general setting of category theory, an isomorphism is a morphism f: XY in a category for which there exists an "inverse" f −1: YX, with the property that both f −1f = idX and f f −1 = idY.

Informally, an isomorphism is a kind of mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, we call the two structures isomorphic. In a certain sense, isomorphic structures are structurally identical, if you choose to ignore finer-grained differences that may arise from how they are defined.

Contents

Purpose

Isomorphisms are studied in mathematics in order to extend insights from one phenomenon to others: if two objects are isomorphic, then any property which is preserved by an isomorphism and which is true of one of the objects, is also true of the other. If an isomorphism can be found from a relatively unknown part of mathematics into some well studied division of mathematics, where many theorems are already proved, and many methods are already available to find answers, then the function can be used to map whole problems out of unfamiliar territory over to "solid ground" where the problem is easier to understand and work with.

Practical example

The following are examples of isomorphisms from ordinary algebra.

  • Consider the logarithm function: For any fixed base b, the logarithm function logb maps from the positive real numbers R+ onto the real numbers R; formally:
    \log_b : \bold{R}^+ \to \bold{R}.
    This mapping is one-to-one and onto, that is, it is a bijection from the domain to the codomain of the logarithm function. In addition to being an isomorphism of sets, the logarithm function also preserves certain operations. Specifically, consider the group (R+,×) of positive real numbers under ordinary multiplication. The logarithm function obeys the following identity:
    \log_b(xy) = \log_b(x) + \log_b(y). \!
    But the real numbers under addition also form a group. So the logarithm function is in fact a group isomorphism from the group (R+,×) to the group (R,+). Logarithms can therefore be used to simplify multiplication of real numbers. By working with logarithms, multiplication of positive real numbers is replaced by addition of logs. This way it is possible to multiply real numbers using a ruler and a table of logarithms, or using a slide rule with a logarithmic scale.
  • Consider the group Z6, the integers from 0 to 5 with addition modulo 6. Also consider the group Z2 × Z3, the ordered pairs where the x coordinates can be 0 or 1, and the y coordinates can be 0, 1, or 2, where addition in the x-coordinate is modulo 2 and addition in the y-coordinate is modulo 3. These structures are isomorphic under addition, if you identify them using the following scheme:
    (0,0) → 0
    (1,1) → 1
    (0,2) → 2
    (1,0) → 3
    (0,1) → 4
    (1,2) → 5
    or in general (a,b) → ( 3a + 4 b ) mod 6. For example note that (1,1) + (1,0) = (0,1) which translates in the other system as 1 + 3 = 4. Even though these two groups "look" different in that the sets contain different elements, they are indeed isomorphic: their structures are exactly the same. More generally, the direct product of two cyclic groups Zm and Zn is isomorphic to Zmn if and only if m and n are coprime.

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