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The Isomorphism Theorems

Although it is not evident at first, factor groups correspond exactly to homomorphic images, and we can use factor groups to study homomorphisms.

The Isomorphism Theorems

Although it is not evident at first, factor groups correspond exactly to homomorphic images, and we can use factor groups to study homomorphisms. We already know that with every group homomorphism \(\phi: G \rightarrow H\) we can associate a normal subgroup of \(G\), \(\ker \phi\). The converse is also true; that is, every normal subgroup of a group \(G\) gives rise to homomorphism of groups.

Let \(H\) be a normal subgroup of \(G\). Define the natural or canonical homomorphism \[\begin{aligned}\end{aligned}\] by \[\begin{aligned}\end{aligned}\]. This is indeed a homomorphism, since \[\begin{aligned}\end{aligned}\]. The kernel of this homomorphism is \(H\). The following theorems describe the relationships between group homomorphisms, normal subgroups, and factor groups.

Theorem: First Isomorphism Theorem

If \(\psi : G \rightarrow H\) is a group homomorphism with \(K =\ker \psi\), then \(K\) is normal in \(G\). Let \(\phi: G \rightarrow G/K\) be the canonical homomorphism. Then there exists a unique isomorphism \(\eta: G/K \rightarrow \psi(G)\) such that \(\psi = \eta \phi\).

Proof

We already know that \(K\) is normal in \(G\). Define \(\eta: G/K \rightarrow \psi(G)\) by \(\eta(gK) = \psi(g)\). We first show that \(\eta\) is a well-defined map. If \(g_1 K =g_2 K\), then for some \(k \in K\), \(g_1 k=g_2\); consequently, \[\begin{aligned}\end{aligned}\]. Thus, \(\eta\) does not depend on the choice of coset representatives and the map \(\eta: G/K \rightarrow \psi(G)\) is uniquely defined since \(\psi = \eta \phi\). We must also show that \(\eta\) is a homomorphism. Indeed, \[\begin{aligned}\eta( g_1K g_2K ) & = \eta(g_1 g_2K) \\ & = \psi(g_1 g_2) \\ & = \psi(g_1) \psi(g_2) \\ & = \eta( g_1K) \eta( g_2K )\end{aligned}\]. Clearly, \(\eta\) is onto \(\psi( G)\). To show that \(\eta\) is one-to-one, suppose that \(\eta(g_1 K) = \eta(g_2 K)\). Then \(\psi(g_1) = \psi(g_2)\). This implies that \(\psi( g_1^{-1} g_2 ) = e\), or \(g_1^{-1} g_2\) is in the kernel of \(\psi\); hence, \(g_1^{-1} g_2K = K\); that is, \(g_1K =g_2K\).

Mathematicians often use diagrams called commutative diagrams to describe such theorems. The following diagram commutes since \(\psi = \eta \phi\).

Notice that in the course of the proof of , we have also proved the following theorem.

Condensed — the full section is in Judson, Abstract Algebra: Theory and Applications.

Symbols used here

x \in A,\ A \subseteq B

element of, subset
x belongs to A; every element of A is in B.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

\blacksquare\ \text{or}\ \square

end of proof (halmos)
Marks the point where the statement has been established.

a \equiv b \pmod n

congruent modulo n
n divides a − b; a and b have the same remainder.

a \mid b,\ \gcd(a,b)

divides, greatest common divisor
b is a multiple of a; the largest number dividing both.

(G, \cdot),\ e,\ g^{-1}

group, identity, inverse
A set with an operation; the do-nothing element; the element that undoes g.

G \cong H,\ G / N

isomorphic, quotient group
Same structure; the group of cosets of a normal subgroup N.

\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n

integers modulo n
The remainders 0…n−1 with clock arithmetic.

\operatorname{Hom}(A, B),\ f \circ g

arrows from A to B, composition
The set of morphisms; do g then f.

Questions people ask

What is a group, in plain words?

A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group — that is where the idea came from.

What is the difference between a ring and a field?

A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.

Wárá

Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Condensed and re-explained here; errors are ours.

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