Fundamental Homomorphism Theorems

The Fundamental Homomorphism Theorem consists of four parts, namely, the First Isomorphism Theorem states that any group is isomorphic to its image modulo its kernel, the Second Isomorphism Theorem gives an isomorphism concerning the product of subgroups and intersection of subgroups, the Third Isomorphism Theorem gives a relationship between normal subgroups and subgroups of the quotient group, and finally, the Fourth Isomorphism Theorem states a sufficient condition for a group to be isomorphic to an internal direct product of normal subgroups. This topic proves the isomorphism theorems and details examples. Also explained is how the First Isomorphism Theorem yields the fact that each homomorphic image is a quotient group.

The following theorems entitled First Isomorphism Theorem, Second Isomorphism Theorem, Third Isomorphism Theorem, and Fourth Isomorphism Theorem are sometimes collectively known as the Fundamental Homomorpishm Theorem. These theorems are the first steps in classifiying the stucture of groups.

Proposition (First Isomorphism Theorem) Let and be groups, and let be a homomorphism from onto with Then the mapping defined by for each is an isomorphism of onto

Proof. To show that is well-defined let Then for some so Thus, implies and so is a mapping. Since for all preserves the group operations. Clearly, is onto because is onto To show that is one-to-one we will show that If then and so Therefore, as desired.

The natural homomorphism shows that each quotient group is the homomorphic image of and the First Isomorphism Theorem states that the converse is also true; that is, each homomorphic image of is a quotient group. Indeed, if a group homomorphism is not onto then you can replace by and thus we have, in the following commutative diagram.

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Example (First Isomorphism Theorem) For let and denote the congruence classes determined by in and respectively. Define by Then is well-defined because if then and therefore, and Since is a homomorphism. Clearly, is onto and since the First Isomorphism Theorem yields

Example (First Isomorphism Theorem) Consider the multiplicative group of nonzero complex numbers. Let be the set of all complex numbers of absolute value Then is a normal subgroup of and the quotient group of is isomorphic to the multiplicative group of all positive real numbers. To see this, use the First Homomorpism Theorem applied to the onto homomorphism defined by with

The following diagram is a mnemonic device for the Second Isomorphism Theorem.
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Proposition (Second Isomorphism Theorem) Let be a group, let be a normal subgroup of and let be any subgroup of Then is a subgroup of is a normal subgroup of and

Proof. Define by for all Then is a homomorphism since Therefore, the image of is a subgroup of and its inverse image in under the natural projection of onto is precisley which must be a subgroup. Since is a normal subgroup of the First Fundamental Theorem yields

How are the subgroups of related to the subgroups of If is a subgroup containing then is certainly a subgroup of in fact, is a normal subgroup of since for all implies for all Therefore, is a group and the elements of are the cosets with Clearly, every such coset is also a coset in Therefore, is a subgroup of The Third Isomorphism Theorem states the case for when and are normal subgroups of The following commutative diagram is a mnemonic device for the Third Isomorphism Theorem.

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Proposition (Third Isomorphism Theorem) Let be a group wth normal subgroups and such that Then is a normal subgroup of and

Proof. Define by for all Then is well-defined since if for some then Since this implies and so It is clear that maps onto Since φ is a homomorphism and since is a normal subgroup of By the First Homomorphism Theorem, as desired.

Proposition (Fourth Isomorphism Theorem) Let be a group with normal subgroups and such that and Then

Proof. The mapping defined by for all is a homomorphism since implies where the last equality comes from for any and which holds because and since and are normal. Clearly, since is onto. Finally, the kernel is trival since if then and so which means that as desired.