No nontrivial homomorphism from quotient group not implies characteristic

From Groupprops
Jump to: navigation, search
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 having no nontrivial homomorphism from its quotient group) 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 more facts about normal subgroup having no nontrivial homomorphism from its quotient group|Get more facts about characteristic subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property normal subgroup having no nontrivial homomorphism from its quotient group but not characteristic subgroup|View examples of subgroups satisfying property normal subgroup having no nontrivial homomorphism from its quotient group and characteristic subgroup

Statement

It is possible to have a group G and a subgroup H such that H is a normal subgroup having no nontrivial homomorphism from its quotient group (i.e., there is no nontrivial homomorphism from G/H to H) but is not a characteristic subgroup of G.

Proof

Infinite abelian example

Let G be the additive group of rational numbers \mathbb{Q} and H be the subgroup \mathbb{Z}. Then:

  • There is no nontrivial homomorphism from G/H to H: This is because every element of G/H has finite order, and no non-identity element of H has finite order.
  • H is not characteristic in G: The automorphism that sends every element to its half in \mathbb{Q} does not preserve H.

Finite example

Let G be direct product of SL(2,3) and Z2 and H be the second direct factor cyclic group:Z2. Then: