Isomorphism - Practical Examples

Practical Examples

The following are examples of isomorphisms from ordinary algebra.

• Consider the logarithm function: For any fixed base b, the logarithm function log_b maps from the positive real numbers R+ onto the real numbers R; formally:
  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:
  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 positive 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 + 4b) 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 Z_m and Z_n is isomorphic to Z_mn if and only if m and n are coprime.