Groupprops, The Group Properties Wiki (pre-alpha)

From Groupprops

Contents 1 Statement 1.1 Verbal statement 1.2 Symbolic statement 2 Definitions used 2.1 Kernel of a group homomorphism 2.2 Normal subgroup 3 Related facts 4 Proof 4.1 Kernel of homomorphism implies normal subgroup 4.2 Normal subgroup implies kernel of homomorphism 5 References 5.1 Textbook references

This article gives the statement, and possibly proof, of a basic fact in group theory.
View a complete list of basic facts in group theory
VIEW FACTS USING THIS: directly | directly or indirectly, upto two steps | directly or indirectly, upto three steps|

This article gives a proof/explanation of the equivalence of multiple definitions for the term normal subgroup
View a complete list of pages giving proofs of equivalence of definitions

Statement

Verbal statement

A subgroup of a group occurs as the kernel of a group homomorphism if and only if it is normal.

Symbolic statement

A subgroup N of a group G occurs as the kernel of a group homomorphism if and only if, for every g in G, .

Definitions used

Kernel of a group homomorphism

A map is a homomorphism of groups if

• for all g,h in G
• 
• 

The kernel of is defined as the inverse image of the identity element under φ.

Normal subgroup

For the purpose of this statement, we use the following definition of normality: a subgroup H is normal in a group G if H contains each of its conjugate subgroups, that is, for every g in G.

Related facts

Closely related to this are the isomorphism theorems.

• First isomorphism theorem
• Second isomorphism theorem
• Third isomorphism theorem

Proof

Kernel of homomorphism implies normal subgroup

Let be a homomorphism of groups. We first prove that the kernel (which we call N) of φ is a subgroup:

• Identity element: Since , e is contained in N
• Product: Suppose a,b are in N. Then and . Using the fact that , we conclude that . Hence ab is also in N.
• Inverse: Suppose a is in N. Then . Using the fact that , we conclude that . Hence, a ^{− 1} is also in N.

Now we need to prove that N is normal. In other words, we must show that if g is in G and n is in N, then gng ^{− 1} is in N.

Since n is in N, φ(n) = e.

Consider . Hence, gng ^{− 1} must belong to N.

Normal subgroup implies kernel of homomorphism

Let N be a normal subgroup of a group G. Then, N occurs as the kernel of a group homomorphism. This group homomorphism is the quotient map , where G / N is the set of cosets of N in G.

The map is defined as follows:

Notice that the map is a group homomorphism if we equip the coset space G / N with the following structure:

(aN)(bN) = abN

This gives a well-defined group structure because, on account of N being normal, the equivalence relation of being in the same coset of N yields a congruence.

Explicitly:

1. The map is well-defined, because if a' = an₁,b' = bn₂ for , then (basically, we're using that bN = Nb).
2. The image of the map can be thought of as a group because it satisfies associativity (((aN)(bN))(cN) = (aN)((bN)(cN))), has an identity element (N itself), has inverses (the inverse of aN is a ^{− 1}N)

Further information: quotient map

References

Textbook references

• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 82, Proposition 7