Fully characteristic subgroups

In summary, the conversation discussed the center of the group G, denoted as Z(G), which is shown to be equal to Z_2. It was mentioned that in order to prove that Z(G) is not a fully characteristic subgroup of G, an endomorphism g needs to be found such that g(Z_2) is not contained in Z_2. However, the method for finding this endomorphism was not clear. It was also mentioned that in order to prove that every fully characteristic subgroup H is also characteristic, it is necessary to show that for every automorphism p in Aut(G), p(H) is contained in H. However, the question of why H is contained in p(H) was raised. One suggestion
  • #1
math8
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Let G=Z_2XS_3 (Z_2:cyclic group of order 2; S_3: Symmetric group on 3) . Show Center of G, Z(G) is not a fully characteristic (or invariant) subgroup of G.

Apparently, Z(G)=Z_2
I know that I need to show that there exists an endomorphism g from G to G such that g(Z_2) is not contained in Z_2.
But I am not sure how.

Also, to prove that every fully characteristic subgroup H is also characteristic, I now how to show that for every automorphism p in Aut(G), p(H) is contained in H, but for some reason I don't see why H is contained in p(H).
 
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  • #2
There are subgroups of S_3 that are isomorphic to Z_2. Map the center to one of those. For the second question, p^(-1) is also in Aut(G).
 

1. What are fully characteristic subgroups?

Fully characteristic subgroups are subgroups of a group that are invariant under all automorphisms of the group. In other words, they are subgroups that are preserved under all possible permutations of the elements of the original group.

2. How do fully characteristic subgroups differ from other types of subgroups?

Unlike normal subgroups, fully characteristic subgroups are not only invariant under all inner automorphisms, but also under all outer automorphisms. This means that the subgroup remains unchanged even if the elements of the original group are rearranged.

3. What is the significance of fully characteristic subgroups in group theory?

Fully characteristic subgroups are important because they provide a way to study the structure of a group by examining its subgroups. They also have applications in other areas of mathematics, such as algebraic geometry and topology.

4. Can fully characteristic subgroups exist in non-Abelian groups?

Yes, fully characteristic subgroups can exist in both Abelian and non-Abelian groups. However, the number of fully characteristic subgroups in a non-Abelian group is usually greater than in an Abelian group.

5. How can fully characteristic subgroups be used in group presentations?

Fully characteristic subgroups can be used to simplify group presentations by reducing the number of generators and relations needed to describe a group. This can make it easier to understand and study the properties of the group.

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