Mathematics Foundations/7.3 Homomorphisms and Isomorphisms

Homomorphisms and Isomorphisms 7.3 Homomorphisms and Isomorphisms Introduction Having explored groups, rings, and fields in the previous sections, we now turn our attention to the mappings between these algebraic structures. Homomorphisms and isomorphisms are fundamental concepts in abstract algebra that allow us to study relationships between different algebraic structures and identify when two seemingly different structures are essentially the same. 7.3.1 Group Homomorphisms Definition and Basic Properties A group homomorphism is a function between two groups and that preserves the group operation: For all , Properties of Group Homomorphisms: , where and are the identity elements of and , respectively. for all . for all and . If is a subgroup of , then is a subgroup of . If is a subgroup of , then is a subgroup of . Kernel and Image The kernel of a group homomorphism , denoted , is the set of elements in that map to the identity element in : The image of a group homomorphism , denoted or , is the set of elements in that are mapped to by at least one element in : Properties: is a normal subgroup of . is a subgroup of . is injective if and only if . The First Isomorphism Theorem for Groups If is a group homomorphism, then: That is, the quotient group is isomorphic to the image of . Examples of Group Homomorphisms The determinant function is a group homomorphism from the general linear group to the multiplicative group of non-zero real numbers. The exponential map defined by is a group homomorphism. The natural projection where is a normal subgroup of is a surjective group homomorphism. 7.3.2 Ring Homomorphisms Definition and Basic Properties A ring homomorphism is a function between two rings and that preserves both ring operations: for all for all If and are rings with unity (multiplicative identity elements and , respectively), we often require that . Properties of Ring Homomorphisms: , where and are the additive identity elements. for all . If is a ring with unity and , then is called a unital ring homomorphism. If is a subring of , then is a subring of . If is a subring of , then is a subring of . Kernel and Image The kernel of a ring homomorphism , denoted , is the set of elements in that map to the additive identity element in : The image of a ring homomorphism , denoted or , is the set of elements in that are mapped to by at least one element in : Properties: is an ideal of . is a subring of . is injective if and only if . The First Isomorphism Theorem for Rings If is a ring homomorphism, then: That is, the quotient ring is isomorphic to the image of . Examples of Ring Homomorphisms The canonical projection defined by is a surjective ring homomorphism with kernel . The inclusion map is an injective ring homomorphism. The evaluation homomorphism defined by is a surjective ring homomorphism. 7.3.3 Field Homomorphisms Definition and Properties A field homomorphism is a ring homomorphism between fields. Due to the special properties of fields, field homomorphisms have additional characteristics: Every non-zero field homomorphism is injective (has trivial kernel). If is a field homomorphism, then is a subfield of . If and are fields of the same characteristic, and is algebraically closed, then any field homomorphism is either injective or constant. Examples of Field Homomorphisms The identity map is a field homomorphism for any field . The inclusion map is a field homomorphism. The complex conjugation is a field automorphism of . 7.3.4 Isomorphisms Definition An isomorphism is a homomorphism that is both injective (one-to-one) and surjective (onto). Two algebraic structures are isomorphic if there exists an isomorphism between them. A group isomorphism is a bijective group homomorphism. A ring isomorphism is a bijective ring homomorphism. A field isomorphism is a bijective field homomorphism. If two structures and are isomorphic, we write . Significance of Isomorphisms Isomorphic structures are essentially the same from an algebraic perspective, differing only in how their elements are labeled or represented. Isomorphisms allow us to: Transfer properties from one structure to another. Classify algebraic structures up to isomorphism. Simplify problems by working with a more convenient isomorphic structure. Examples of Isomorphisms The group of positive real numbers under multiplication is isomorphic to via the exponential function . The ring is isomorphic to the ring of Gaussian integers . The field is isomorphic to . 7.3.5 Automorphisms Definition and Properties An automorphism is an isomorphism from an algebraic structure to itself. The set of all automorphisms of a structure forms a group under composition, denoted . Inner Automorphisms In a group , for each element , the map defined by is an automorphism called an inner automorphism. The set of all inner automorphisms forms a normal subgroup of , denoted . The quotient group is called the outer automorphism group of , denoted . Galois Theory A major application of automorphisms is in Galois theory, where the automorphism group of a field extension plays a central role. If is a field extension, the group of automorphisms of that fix pointwise is called the Galois group of the extension, denoted . The fundamental theorem of Galois theory establishes a one-to-one correspondence between the subgroups of and the intermediate fields between and (when is a Galois extension). Examples of Automorphisms The automorphism group of over consists of the identity map and complex conjugation. The automorphism group of the quaternions is isomorphic to , the group of rotations in three-dimensional space. The automorphism group of the field is the Klein four-group . 7.3.6 Endomorphisms Definition and Properties An endomorphism is a homomorphism from an algebraic structure to itself. The set of all endomorphisms of a structure forms a monoid under composition, denoted . In the case of vector spaces, endomorphisms are precisely the linear transformations from the space to itself, and they form an algebra over the base field. Examples of Endomorphisms The map defined by is a group endomorphism of . The zero map defined by is an endomorphism of any vector space . The Frobenius endomorphism defined by where is a field of characteristic . 7.3.7 Applications of Homomorphisms and Isomorphisms Group Theory: Cayley's theorem states that every group is isomorphic to a subgroup of a symmetric group. Ring Theory: The Chinese remainder theorem can be stated in terms of ring isomorphisms. Representation Theory: Group representations are homomorphisms from a group to a group of matrices. Galois Theory: The fundamental theorem of Galois theory establishes a correspondence between subfields and subgroups of the Galois group. Cryptography: Many cryptographic protocols rely on the difficulty of finding isomorphisms between certain algebraic structures. Category Theory: Homomorphisms are the morphisms in the categories of groups, rings, and other algebraic structures. 7.3.8 Homomorphism Theorems First Isomorphism Theorem We have already seen the First Isomorphism Theorem for groups and rings. In general, if is a homomorphism of algebraic structures, then . Second Isomorphism Theorem For Groups: If is a subgroup of a group and is a normal subgroup of , then is a normal subgroup of and . For Rings: If is a subring of a ring and is an ideal of , then is an ideal of and . Third Isomorphism Theorem For Groups: If and are normal subgroups of a group with , then is a normal subgroup of and . For Rings: If and are ideals of a ring with , then is an ideal of and . Correspondence Theorem For Groups: If is a normal subgroup of a group , then there is a one-to-one correspondence between the subgroups of and the subgroups of containing . For Rings: If is an ideal of a ring , then there is a one-to-one correspondence between the ideals of and the ideals of containing . Exercises 1. Prove that if is a group homomorphism, then , where and are the identity elements of and , respectively. 2. Let be defined by (the congruence class of modulo ). Show that is a surjective ring homomorphism and determine its kernel. 3. Prove that the additive group is isomorphic to the multiplicative group of positive real numbers. 4. Show that the rings and are isomorphic. 5. Determine all ring homomorphisms from to . 6. Prove that any homomorphism from a field to a non-zero ring is injective. 7. Find the automorphism group of the field . 8. Show that the only ring automorphism of is the identity map. 9. Prove that if is a simple group, then any non-trivial homomorphism is either injective or maps all of to the identity element of . 10. Let be a ring homomorphism. Prove that if is an ideal of , then is an ideal of . 11. If is a group and is a subgroup of index 2, prove that is a normal subgroup of . 12. Determine the kernel and image of the homomorphism defined by .