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From Groupprops

This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
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Contents 1 Statement 1.1 Verbal statement 1.2 Symbolic statement 1.3 Property-theoretic statement 2 Definitions used 2.1 Normal subgroup 2.2 Transfer condition 3 Related facts 3.1 Further facts 3.2 Related metaproperties satisfied by normality 3.3 Transfer condition for other subgroup properties 3.4 Analogues in other algebraic structure 4 Proof 4.1 Hands-on proof 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
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Statement

Verbal statement

If a subgroup is normal in the group, its intersection with any other subgroup is normal in that subgroup.

Symbolic statement

Let be a normal subgroup and let K be any subgroup of G. Then, .

Property-theoretic statement

The subgroup property of being normal satisfies the transfer condition.

Definitions used

Normal subgroup

A subgroup H of a group G is said to be normal if for any and , .

Transfer condition

A subgroup property p is said to satisfy transfer condition if whenever H,K are subgroups of G and H has property p in G, has property p in K.

Related facts

Further facts

• Second isomorphism theorem: This result equates the quotient of the non-normal subgroup, by the intersection, with the quotient of the product of subgroups, by the normal subgroup. Specifically, it states that if H is normal in G and K is any subgroup of G, we have .

Related metaproperties satisfied by normality

• Normality satisfies intermediate subgroup condition: The intermediate subgroup condition is weaker. It says that if are subgroups are H is normal in G, then H is normal in K.
• Normality satisfies inverse image condition: The inverse image condition is stronger. It says that the inverse image of a normal subgroup under a homomorphism is normal.

Other metaproperties satisfied by normality, that are somewhat related:

• Normality satisfies image condition: The image of a normal subgroup under a surjective homomorphism is normal in the image.
• Normality is upper join-closed

Transfer condition for other subgroup properties

• Subnormality satisfies transfer condition: This follows directly from the fact that normality satisfies the transfer condition, and the fact that transfer condition is subordination-closed.
• Permutability satisfies transfer condition

Analogues in other algebraic structure

• Ideal property satisfies transfer condition in Lie rings

Proof

Hands-on proof

Given: A group G, a normal subgroup and a subgroup

To prove: . In other words, we need to prove that given any and , .

Proof: Since , we in particular have . Since (viz H is normal in G), .

But we also have that and . Since K is a subgroup, .

Combining these two facts, .

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 88, Exercise 24
• Topics in Algebra by I. N. Herstein, ^{More info}, Page 53, Problem 5