First isomorphism theorem

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This article gives the statement, and possibly proof, of a basic fact in group theory.
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This article is about an isomorphism theorem in group theory.
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Name

This result is termed the first isomorphism theorem, or sometimes the fundamental theorem of homomorphisms.

Statement

General version

Let G be a group and \varphi:G \to H be a homomorphism of groups. The first isomorphism theorem states that the kernel of \varphi is a Normal subgroup (?), say N, and there is a natural isomorphism:

G/N \cong \varphi(G)

where \varphi(G) denotes the image in H of G under \varphi.

More explicitly, if \alpha:G \to G/N is the quotient map, then there is a unique isomorphism \psi:G/N \to \varphi(G) such that \psi \circ \alpha = \varphi.

Version for surjective homomorphism

This is a special case of the more general statement.

Let G be a group and \varphi:G \to H be a surjective homomorphism of groups. Then, if N is the kernel of \varphi, we have:

G/N \cong H

More explicitly, if \alpha:G \to G/N is the quotient map, then there is a unique isomorphism \psi:G/N \to \varphi(G) such that \psi \circ \alpha = \varphi.

Universal algebraic statement

  • In the variety of groups, every ideal (normal subgroup) is a kernel
  • In the variety of groups, a congruence is completely determined by its kernel. In other words, simply knowing the inverse image of the identity element for a surjective homomorphism, determines the nature of the homomorphism.

This is encoded by saying that the variety of groups is ideal-determined.

Related facts

Related facts about groups

Facts used

  1. Normal subgroup equals kernel of homomorphism: Given any homomorphism \varphi:G \to H of groups, the kernel of \varphi (i.e., the inverse image of the identity element) is a normal subgroup of G. Further, given any normal subgroup N of G, there is a natural quotient group G/N.

Proof

Given: A homomorphism of groups \varphi:G \to H, with kernel N (i.e. N is the inverse image of the identity element).

To prove: N is a normal subgroup, and G/N \cong \varphi(G)

Proof: Two steps of the proof are done at normal subgroup equals kernel of homomorphism (fact (1)):

  1. The kernel of any homomorphism is a normal subgroup
  2. If N is a normal subgroup, we can define a quotient group G/N which is the set of cosets of N, with multiplication of cosets given by:

(aN)(bN) = (ab)N

It now remains to show that we can identify G/N isomorphically with \varphi(G). Consider the map from G/N to \varphi(G):

\overline{\varphi}: aN \mapsto \varphi(a)

We first argue that this map is well-defined. For this, observe that if b = an, n \in N, then:

\varphi(b) = \varphi(an) = \varphi(a)\varphi(n) = \varphi(a)

In other words, any two elements in the same coset of N in G get mapped to the same element of H.

Next, we argue that the map is a group homomorphism. Indeed, if aN and bN are two cosets, then:

\varphi(ab) = \varphi(a)\varphi(b) \implies \overline{\varphi}(abN) = \overline{\varphi}(aN)\overline{\varphi}(bN)

(similar checks work for identity element and inverses).

Next, we argue that the map is injective. Indeed, if aN is sent to the identity element, then \varphi(a) = e, forcing a \in N.

Finally, we argue that the map is surjective. By definition, any element in \varphi(G) can be written as \varphi(a) for some a \in G, and hence occurs as \overline{\varphi}(aN).

References

Textbook references

  • Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, More info, Page 68-69, Theorem (10.9)
  • Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, More info, Page 97 (Theorem 16): the proof is spread across previous sections, and is not given after the statement