__Example of Factor Group Construction__

Let **Z** stand for the set of all integers; **5Z** stand for all multiples of five; and **15Z** for all multiples of fifteen.

The set **Z** along with the binary operation of addition forms a group. The proper notation is **[Z, +]** but it's commonly understood by **Z** alone.

We can also have groups composed of other things, like maps and matrices, dimensional hierarchies (homology groups) and geometric shapes. With the latter, we're looking for closed sub-symmetries embedded in larger symmetries.

The set **5Z** under the operation of addition is a normal subgroup of **Z** as well as the identity equivalence class of the factor group -- **Z/5Z**. As a group itself it also has subgroups of its own -- **15Z** -- for instance.

If we imagine all multiples of five spread out on a table, and below them all multiples of fifteen, we could draw lines from entries in the set of **5Z** to corresponding ones in the set of **15Z**. The numbers in the set of **5Z** left out of the mapping process can then be organized into respective classes according to the rules of arithmetic and the axioms of groups. By so doing we generate the factor group -- **5Z/15Z** which is isomorphic to **Z _{3}**. By the way,

**Z _{3}** is the group composed of the numbers {0,1,2} under the operation addition. The elements of

They look like: (..., -30, -15, 0, 15, 30, ...) the identity class which maps to zero in **Z _{3}**, (..., -40, -25, -10, 5, 20, 35, ...) which maps to

Addition is by representatives from a class.

For example: (-10+30=20), (-25+35=10), (-40+25=-15), and (-20+10=-10). Analogous computations in **Z _{3}** are, respectively: 1+0=1, 1+1=2, 1+2=0, and 2+2=1.

And the addition of *any* representatives from these classes will result in the same class.

For example: (-10+30-20=0) falls into the class identified with **0** in **Z _{3}**, as do the results of (-25+15-35=-45) and (35+45-20=60). They all correspond to

**15Z** is not only seen to be the identity *element* of the factor group -- **5Z/15Z** -- but also, in the *act* of factoring, takes on the additional role as **kernel** of the homomorphism (structure-preserving map) which assigns numbers to respective sets. The kernel generates the equivalence class composed of multiples of **15** and by so doing establishes how numbers are to be distributed -- by the difference of fifteen -- and partitioned into classes -- the elements of the factor group. The kernel sets the pattern, the stamp, the lay of the land, and the constant of difference, in this particular case.

In a nutshell: This normal subgroup/identity equivalence class/kernel of homomorphism -- **15Z** -- is mapped to zero -- our additive identity element -- in **Z _{3}** under the isomorphism.

And as the factor group -- **5Z/15Z** -- is *a* homomorphic image of **5Z**, the operation of addition remains invariant as does the symmetric relationship among parts -- how it's structured. In this manner you can find all homomorphic images of **5Z** without going *outside* the group itself.

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