Finite cyclic group

Finite cyclic group

By chance , It seems that in all finite cyclic groups , A unit is an endogenetic gift of a group , Without additional assumptions .

This explanation , If there were some laws in the universe , As long as we observe natural numbers , There is no denying that somewhere , It can put the original great , Mapping to minimal , So as to turn the end point into the starting point .

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• Finite cyclic group
• The unitary element in finite cyclic groups
• How to construct such a group

  • Natural number
  • Rational number

The unitary element in finite cyclic groups

First assume a binary operation ,

If a set is closed to this operation , Is called a group .

Further , If all the elements of a set can be represented as In the form of ,

that The generator of a set .

Then something magical happened , In this set , The existence of monoids is inevitable , No additional assumptions are required .

More specifically ,

If the generator of a group is unique , And its elements are finite , Then the accumulation of all elements is the unit of this group .

The proof process is extremely simple , We can refer to the proof method of Fermat's small theorem .

prove ：

First , Suppose there are Elements ,

We don't know its specific value , But it's definitely a value , For the time being Express .

From the properties of the generator ,

therefore , All elements can be considered

They are related to each other Carry out operations , Get new set

Accumulate all elements of the new set

namely , Is the unit of a set .

Prove completion .

How to construct such a group

Since this kind of finite cyclic group has such good properties , So how do we construct it ？

Subsets of rational numbers can construct such groups .

Natural number

On a small scale , Subsets of natural numbers are more intuitive examples , here , Any natural number can be used as a generator . Because the directly constructed group is a countable infinite group , In order to construct finite groups , Take the remainder of all elements . This is also the idea of Fermat's theorem .

among , For any natural number , Prime number . It's not hard to see. , The unit of this set is , Prove the following

Prove completion .

Rational number

However , If you do not need to take the remainder , Only any natural number is considered , Construct such a sequence

It is obviously not something meaningful , But with a little change

There is a new sequence

The length of the left sequence is This is to enable the results of pairing operations on all elements on the right to find the corresponding elements on the left , So as to satisfy the closeness of the group .

But here we find , In the original sequence It can be removed , Its value is actually a result

in other words , No matter what What is the value of , We can all find the unit from the generated group . therefore , Although almost all sets need to start from 1 Start talking about , but 1 The existence of is a mirror that precedes the assembly .

Of course, this is not nothing , Because as long as you take the logarithm of this set , You can generate a sequence of natural numbers . in other words , If there were some laws in the universe , As long as we observe natural numbers , There is no denying that somewhere , It can put the original great , Mapping to minimal

So as to turn the end point into the starting point .