Cyclic groups

#matematik-groups

Definition: Cyclic group

Let be a group and . Then is the smallest subgroup of containing , called the subgroup generated by .

If for some , we write and call it the cyclic group generated by .

Theorem

Let be a group and . Then Proof:
Let be the right hand expression above. By the construction of , we know . We also know that , so . For any such that it is clear that since each element of is in and thus all compositions of elements of . Thus , so .

Remark

Cyclic groups are abelian since .

Example

is called the free cyclic group. For any we also have and If we get which is called the cyclic group of order .

We call them the free cyclic group and cyclic groups of order because we can show that every cyclic group is isomorphic to one of these. For that we first need a few lemmas.

Lemma

If is a group, is cyclic if and only if there exists a surjective .

Proof:
: We have .
: Let , and we get , which combined with the fact that is surjective means that .

Lemma

If , then for some .

Proof:
If , then . Otherwise, let (which exists since if then ). Then we clearly have . Finally, let . Then we have for some , which means , implying that by minimality, so .

Theorem

Let be a cyclic group. Then

  1. If , .
  2. If , .

Proof:
Since is syclic there exists a surjective . The universal property gives a bijection . We know for some , so where in the first case, , and in the second, .

Theorem

Let be a cyclic group and . Then is cyclic.

Proof:
By Lemma, there exists a surjective homomorphism , so . By Lemma, there exists an such that . If , then , and otherwise, which by Lemma means that is cyclic.

Definition: Order of element

Let be a group and . The order of is

This means that if , and .

Theorem: Fermat's Little Theorem

Let be a finite group and . Then Proof:
The divisibility follows from Lagrange's Theorem, and the second part follows from the fact that .

Corollary: Fermat's Little Theorem, classic version

Let be prime and . Then Proof:
Let (recall that '' denotes exclusion of zero). We first show that is a group with respect to multiplication. Multiplication is well defined as since divides neither nor , so does not divide either since is prime, meaning , so . Furthermore, this multiplication is injective since if , then , so meaning since . This means , so . Since is clearly a unit element, it follows that inverse elements exist, hence is a group under multiplication.

Now, if , then . Otherwise, we have , which by the group theoretic version of Fermat's Little Theorem means that

Corollary

Let be a finite group with order , where is prime. Then

  1. ,
  2. For all , , and .

Proof:
b) Since , we know since is prime and . Hence , so .
a) From Theorem, we know .