Abstract Algebra/Group Theory/Homomorphism

We are finally making our way into the meat of the theory. In this section we will study structure-preserving maps between groups. This study will open new doors and provide us with a multitude of new theorems.

Up until now we have studied groups at the "element level". Since we are now about to take a step back and study groups at the "homomorphism level", readers should expect a sudden increase in abstraction starting from this section. We will try to ease the reader into this increase by keeping one foot at the "element level" throughout this section.

From here on out the notation will denote the identity element in the group unless otherwise specified.

Group homomorphismsEdit

Definition 1: Let   and   be groups. A homomorphism from   to   is a function   such that for all  ,

 .

Thus, a homomorphism preserves the group structure. We have included the multiplication symbols here to make explicit that multiplication on the left side occurs in  , and multiplication on the right side occurs in  .

Already we see that this section is different from the previous ones. Up until now we have, excluding subgroups, only dealt with one group at a time. No more! Let us start by deriving some elementary and immediate consequences of the definition.

Theorem 2: Let   be groups and   a homomorphism. Then  . In other words, the identity is mapped to the identity.

Proof: Let  . Then,  , implying that   is the identity in  , proving the theorem.

Theorem 3: Let   be groups and   a homomorphism. Then for any  ,  . In other words, inverses are mapped to inverses.

Proof: Let  . Then   implying that  , as was to be shown.

Theorem 4: Let   be groups,   a homomorphism and let   be a subgroup of  . Then   is a subgroup of  .

Proof: Let  . Then   and  . Since  ,  , and so   is a subgroup of  .

Theorem 5: Let   be groups,   a homomorphism and let   be a subgroup of  . Then   is a subgroup of  .

Proof: Let  . Then  , and since   is a subgroup,  . But then,  , and so   is a subgroup of  .

From Theorem 4 and Theorem 5 we see that homomorphisms preserve subgroups. Thus we can expect to learn a lot about the subgroup structure of a group   by finding suitable homomorphisms into  .

In particular, every homomorphism   has associated with it two important subgroups.

Definition 6: A homomorphism is called an isomorphism if it is bijective and its inverse is a homomorphism. Two groups are called isomorphic if there exists an isomorphism between them, and we write   to denote "  is isomorphic to  ".

Theorem 7: A bijective homomorphism is an isomorphism.

Proof: Let   be groups and let   be a bijective homomorphism. We must show that the inverse   is a homomorphism. Let  . then there exist unique   such that   and  . Then we have   since   is a homomorphism. Now apply   to all equations. We obtain  ,   and  , so   is a homomorphism and thus   is an isomorphism.

Definition 8: Let   be groups. A homomorphism that maps every element in   to   is called a trivial homomorphism (or zero homomorphism), and is denoted by  

Definition 9: Let   be a subgroup of a group  . Then the homomorphism   given by   is called the inclusion of   into  . Let   be a group isomorphic to a subgroup   of a group  . Then the isomorphism   induces an injective homomorphism   given by  , called an imbedding of   into  . Obviously,  .

Definition 10: Let   be groups and   a homomorphism. Then we define the following subgroups:

i)  , called the kernel of  , and
ii)  , called the image of  .

Theorem 11: The composition of homomorphisms is a homomorphism.

Proof: Let   be groups and   and   homomorphisms. Then   is a function. We must show it is a homomorphism. Let  . Then  , so   is indeed a homomorphisms.

Theorem 12: Composition of homomorphisms is associative.

Proof: This is evident since homomorphisms are functions, and composition of functions is associative.

Corollary 13: The composition of isomorphisms is an isomorphism.

Proof: This is evident from Theorem 11 and since the composition of bijections is a bijection.

Theorem 14: Let   be groups and   a homomorphism. Then   is injective if and only if  .

Proof: Assume   and  . Then  , implying that  . But by assumption then  , so   is injective. Assume now that   and  . Then there exists another element   such that  . But then  . Since both   and   map to  ,   is not injective, proving the theorem.

Corollary 15: Inclusions are injective.

Proof: The result is immediate. Since   for all  , we have  .

The kernel can be seen to satisfy a universal property. The following theorem explains this, but it is unusually abstract for an elementary treatment of groups, and the reader should not worry if he/she cannot understand it immediately.

 
Commutative diagram showing the universal property of kernels.

Theorem 16: Let   be groups and   a group homomorphism. Also let   be a group and   a homomorphism such that  . Also let   is the inclusion of   into  . Then there exists a unique homomorphism   such that .

Proof: Since  , by definition we must have  , so   exists. The commutativity   then forces  , so   is unique.

Definition 17: A commutative diagram is a pictorial presentation of a network of functions. Commutativity means that when several routes of function composition from one object lead to the same destination, the two compositions are equal as functions. As an example, the commutative diagram on the right describes the situation in Theorem 16. In the commutative diagrams (or diagrams for short, we will not show diagrams which no not commute) shown in this chapter on groups, all functions are implicitly assumed to be group homomorphisms. Monomorphisms in diagrams are often emphasized by hooked arrows. In addition, epimorphims are often emphasized by double headed arrows. That an inclusion is a monomorphism will be proven shortly.

Remark 18: From the commutative diagram on the right, the kernel can be defined completely without reference to elements. Indeed, Theorem 16 would become the definition, and our Definition 10 i) would become a theorem. We will not entertain this line of thought in this book, but the advanced reader is welcome to work it out for him/herself.

Automorphism GroupsEdit

In this subsection we will take a look at the homomorphisms from a group to itself.

Definition 19: A homomorphism from a group   to itself is called an endomorphism of  . An endomorphism which is also an isomorphism is called an automorphism. The set of all endomorphisms of   is denoted  , while the set of all automorphisms of   is denoted  .

Theorem 20:   is a monoid under composition of homomorphisms. Also,   is a submonoid which is also a group.

Proof: We only have to confirm that   is closed and has an identity, which we know is true. For  , the identity homomorphism   is an isomorphism and the composition of isomorphisms is an isomorphism. Thus   is a submonoid. To show it is a group, note that the inverse of an automorphism is an automorphism, so   is indeed a group.

Groups with OperatorsEdit

An endomorphism of a group can be thought of as a unary operator on that group. This motivates the following definition:

Definition 21: Let   be a group and  . Then the pair   is called a group with operators.   is called the operator domain and its elements are called the homotheties of  . For any  , we introduce the shorthand   for all  . Thus the fact that the homotheties of   are endomorphisms can be expressed thus: for all   and  ,  .

Example 22: For any group  , the pair   is trivially a group with operators.

Lemma 23: Let   be a group with operators. Then   can be extended to a submonoid   of   such that the structure of   is identical to  .

Proof: Let   include the identity endomorphism and let   be a generating set. Then   is closed under compositions and is a monoid. Since any element of   is expressible as a (possibly empty) composition of elements in  , the structures are identical.

In the following, we assume that the operator domain is always a monoid. If it is not, we can extend it to one by Lemma 23.

Definition 24: Let   and   be groups with operators with the same operator domain. Then a homomorphism   is a group homomorphism   such that for all   and  , we have  .

Definition 25: Let   be a group with operators and   a subgroup of  . Then   is called a stable subgroup (or a  -invariant subgroup) if for all   and  ,  . We say that   respects the homotheties of  . In this case   is a sub-group with operators.

Example 26: Let   be a vector space over the field  . If we by   denote the underlying abelian group under addition, then   is a group with operators, where for any   and  , we define  . Then the stable subgroups are precisely the linear subspaces of   (show this).

ProblemsEdit

Problem 1: Show that there is no nontrivial homomorphism from   to  .