S. D. V. Stephens

notes / Studies in Algebra and Group Theory / chapter 1: Sets and Groups / From my notebooks

N5 Subgroups of ℤ, Cyclic Groups and Equivalence Relations

Recall

Definition N5.1 (Subgroup). A subgroup is an such that is a group and is a nonempty subset which is closed under and inverses.

Definition N5.2 (Group homomorphism). Given groups , a homomorphism is a map which takes products to products, , and , .

Definition N5.3 (Kernel and image). The kernel of a group homomorphism is . is a subgroup of (a “normal subgroup” (?)). is injective .

The image of a group homomorphism is This is a subgroup of . is surjective when .

Remark. If is injective, is isomorphic to (by , ).

(Question to self: for a generating set of , do you need both , or even ?)

Subgroups of

Here is simply addition. Given , is a subgroup. This leads to:

Proposition N5.4. All nontrivial subgroups of are of this form.

Proof. Given a nontrivial subgroup , there exists with . Let be the smallest positive element of . Then all the multiples of are in (). Given any , by Euclidean division there exist (the quotient) and (the remainder) such that . Since and , thus . We can’t have (because of how was chosen), so and . Hence , so . ∎

So every subgroup of is generated by a single element. Generation: the element that builds all other elements in the group by repeating the group operation.

The subgroup generated by a set

Question N5.5. Given a subset , what is the smallest subgroup of which contains ? This is denoted and is called the subgroup generated by .

Answer to the question: This is an intersection of subgroups, hence a subgroup. (Confused about this :-|)

A more constructive answer: where the empty product () is . This contains , is a subgroup of , and is therefore the smallest.

Cyclic groups and order

Definition N5.6 (Cyclic group). A group is said to be cyclic if it is generated by a single element.

Example N5.7. and are both generated by . But in , when we take we get the identity (). These are the only cyclic groups “up to isomorphism”.

Example N5.8. Let be any element. Then , is a homomorphism since with image , the cyclic subgroup of generated by .

Definition N5.9 (Order of an element). The order of is the smallest positive such that ; if there is none, is of infinite order. (Not , which is the order of the group!)

If has infinite order then the powers of are all distinct, is injective, and is isomorphic to . If, however, has finite order, then ( = the smallest positive element of the kernel). Also is isomorphic to .

Example N5.10. is isomorphic to via . Observe: has order (the smallest multiple of both and ). Similarly when (Chinese remainder theorem), but since in every nonzero element has order .

Classifying finite groups up to isomorphism?

Increasingly difficult as increases.

  • Order 1:
  • Order 2:
  • Order 3:
  • Order 4: ,

The classification of finite simple groups was completed in the 1980s, but we will learn the key tools and methods. That end goal was finished, thus it is a closed question.

Equivalence relations

(Artin §2.7 and Halmos.)

Definition N5.11 (Equivalence relation). An equivalence relation on a set is a binary relation, i.e. a subset of (the set of tuples with ), such that

  1. reflexive: for all ;
  2. symmetric: ;
  3. transitive: if and , then .

Definition N5.12 (Equivalence class, quotient set). The equivalence class of is , often denoted . Using our axioms, if then .

  • The equivalence classes partition into mutually disjoint subsets.
  • The quotient set is the set of equivalence classes. It comes with the surjective map , .

Example N5.13 (Congruence mod ). Let . Given , set iff divides (congruence mod ). This is an equivalence relation. There are equivalence classes: So . The formal definition of uses equivalence classes. Note: equivalence classes may have many different names.

Lecture 3 continued: maps and quotients

Given a surjective map of sets , we can set iff . This is an equivalence relation.

Note. The equivalence classes are the subsets , , and Since for all , this could also send to ; that may cause trouble in general, but is fine here because .

Generally, given a map and an equivalence relation on , when does induce a map Necessary and sufficient: is compatible with , in the sense that . Then factors through the quotient:

Example N5.14. The addition operation induces an addition operation on , , because it is compatible with the equivalence relation: and the diagonal composite sends . (I’m going to cry.)

Cosets of a kernel

Assume we have a surjective group homomorphism , and , a subgroup of . Then So for the relation , the equivalence class of consists of the : it is the left coset .