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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normal subgroup) need not satisfy the second subgroup property (i.e., characteristic subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications |Get help on looking up subgroup property implications/non-implications
Get more facts about normal subgroup|Get more facts about characteristic subgroup

Contents 1 Statement 2 Related facts 2.1 Converse 2.2 Other related facts 2.3 Related group properties 3 Proof 3.1 Example of a direct product 4 GAP implementation 4.1 Implementation of the generic example 5 References 5.1 Textbook references

Statement

A normal subgroup of a group need not be a characteristic subgroup.

Related facts

Converse

Further information: Characteristic implies normal

The converse statement is indeed true. That is, every characteristic subgroup is normal.

Other related facts

• Normality is not transitive: A normal subgroup of a normal subgroup need not be normal.
• Characteristic of normal implies normal: A characteristic subgroup of a normal subgroup is normal.
• Left transiter of normal is characteristic: If H is a subgroup of K such that whenever K is normal in G, H is normal in G, then H is characteristic in K.
• Direct factor not implies characteristic

Related group properties

There are some groups in which every normal subgroup is characteristic. Further information: Group in which every normal subgroup is characteristic

Proof

Example of a direct product

Let G be any nontrivial group. Then consider , viz., the external direct product of G with itself. The subgroups and are direct factors of K, and are hence both normal in K. Note also that they are distinct, since G is nontrivial.

However, the exchange automorphism:

exchanges the subgroups G₁ and G₂. Thus, neither G₁ nor G₂ is invariant under all the automorphisms, so neither is characteristic. Thus, G₁ and G₂ are both normal subgroups of K that are not characteristic.

Note that this example also shows that direct factor does not imply characteristic subgroup.

GAP implementation

Implementation of the generic example

Before using this generic example, you need to define G for GAP, choosing any nontrivial group (double semicolons have been used here to suppress GAP's output for the first three commands, which depends on the specific choice of G -- you can use single semicolons instead).

gap> K := DirectProduct(G,G);;
gap> G1 := Image(Embedding(K,1));;
gap> G2 := Image(Embedding(K,2));;
gap> IsSubgroup(K,G1);
true
gap> IsSubgroup(K,G2);
true
gap> IsNormal(K,G1);
true
gap> IsNormal(K,G2);
true
gap> IsCharacteristicSubgroup(K,G1);
false
gap> IsCharacteristicSubgroup(K,G2);
false

References

Textbook references

• Topics in Algebra by I. N. Herstein^{More info}, Page 70 (Problem 7(a))
• Abstract Algebra by David S. Dummit and Richard M. Foote, ISBN 0471433349^{More info}, Page 137 (Problem 6)