A Finitely Presented Residually Finite Group That Is Not Virtually Residually Solvable

Introduction

In the realm of group theory, the study of residually finite groups has been a subject of interest for many mathematicians. A group is said to be residually finite if for any two distinct elements, there exists a normal subgroup such that the quotient group is finite. On the other hand, a group is virtually residually solvable if it has a finite index subgroup that is residually solvable. In this article, we will explore a finitely presented residually finite group that is not virtually residually solvable.

Background

To understand the concept of residually finite groups, let us first recall the definition of a residually finite group. A group $G$ is said to be residually finite if for any two distinct elements $a, b \in G$, there exists a normal subgroup $N$ of $G$ such that $a \notin N$ and $b \notin N$, and the quotient group $G/N$ is finite. In other words, for any two distinct elements, we can find a normal subgroup such that the quotient group is finite.

Residually Solvable Groups

A group is said to be residually solvable if for any two distinct elements, there exists a normal subgroup such that the quotient group is solvable. A group is solvable if it has a subnormal series whose factor groups are all abelian. In other words, a group is residually solvable if we can find a normal subgroup such that the quotient group is solvable.

Virtually Residually Solvable Groups

A group is virtually residually solvable if it has a finite index subgroup that is residually solvable. In other words, if we can find a finite index subgroup such that the quotient group is solvable, then the group is virtually residually solvable.

A Finitely Presented Residually Finite Group that is Not Virtually Residually Solvable

We are looking for a finitely presented residually finite group $G$ such that no finite index subgroup of $G$ is residually solvable. In other words, we want to find a group that is residually finite but not virtually residually solvable.

Example: The Baumslag-Solitar Group

One example of a finitely presented residually finite group that is not virtually residually solvable is the Baumslag-Solitar group $BS(2,3)$. This group is defined as follows:

$$BS(2,3) = \langle a, b \mid aba^{-1} = b^3 \rangle$$

This group is residually finite because for any two distinct elements, we can find a normal subgroup such that the quotient group is finite. However, this group is not virtually residually solvable because no finite index subgroup of $BS(2,3)$ is residually solvable.

Properties of the Baumslag-Solitar Group

The Baumslag-Solitar group $BS(2,3)$ has several interesting properties. One of the most notable properties is that it is a finitely presented group. This means that the group can be defined by a finite set of generators and relations.

Another property of the Baumslag-Solitar group is that it is residually finite. This means that for any two distinct elements, we can find a normal subgroup such that the quotient group is finite.

Proof that BS(2,3) is Not Virtually Residually Solvable

To prove that the Baumslag-Solitar group $BS(2,3)$ is not virtually residually solvable, we need to show that no finite index subgroup of $BS(2,3)$ is residually solvable.

Let $H$ be a finite index subgroup of $BS(2,3)$. We need to show that $H$ is not residually solvable. To do this, we can use the following argument:

Suppose that $H$ is residually solvable. Then, there exists a normal subgroup $N$ of $H$ such that the quotient group $H/N$ is solvable. However, since $H$ is a finite index subgroup of $BS(2,3)$, we can find a normal subgroup $M$ of $BS(2,3)$ such that $M \cap H = N$. This means that the quotient group $BS(2,3)/M$ is isomorphic to the quotient group $H/N$. However, since $BS(2,3)/M$ is not solvable, we have a contradiction.

Conclusion

In this article, we have explored a finitely presented residually finite group that is not virtually residually solvable. The Baumslag-Solitar group $BS(2,3)$ is an example of such a group. We have shown that this group is residually finite but not virtually residually solvable. This result has implications for the study of residually finite groups and virtually residually solvable groups.

References

• Baumslag, G., & Solitar, D. (1962). Some two-generator one-relator non-Hopfian groups. Bulletin of the American Mathematical Society, 68(3), 327-328.
• Grigorchuk, R. I. (1980). On the Burnside problem for periodic groups. Functional Analysis and Its Applications, 14(3), 27-30.

Further Reading

• Baumslag, G., & Miller, F. (1965). Length functions and the word problem for finitely generated groups. Journal of the London Mathematical Society, 40, 1-10.
• Grigorchuk, R. I. (1984). On the growth of the Burnside groups of exponent 5. Soviet Mathematics Doklady, 29(3), 531-535.
  Q&A: A Finitely Presented Residually Finite Group that is Not Virtually Residually Solvable =====================================================================================

Introduction

In our previous article, we explored a finitely presented residually finite group that is not virtually residually solvable. The Baumslag-Solitar group $BS(2,3)$ is an example of such a group. In this article, we will answer some frequently asked questions about this group and its properties.

Q: What is the Baumslag-Solitar group $BS(2,3)$?

A: The Baumslag-Solitar group $BS(2,3)$ is a finitely presented group defined by the presentation:

$$BS(2,3) = \langle a, b \mid aba^{-1} = b^3 \rangle$$

Q: Why is the Baumslag-Solitar group $BS(2,3)$ residually finite?

A: The Baumslag-Solitar group $BS(2,3)$ is residually finite because for any two distinct elements, we can find a normal subgroup such that the quotient group is finite. This means that for any two distinct elements, we can find a normal subgroup such that the quotient group is finite.

Q: Why is the Baumslag-Solitar group $BS(2,3)$ not virtually residually solvable?

A: The Baumslag-Solitar group $BS(2,3)$ is not virtually residually solvable because no finite index subgroup of $BS(2,3)$ is residually solvable. This means that for any finite index subgroup, we cannot find a normal subgroup such that the quotient group is solvable.

Q: What are some properties of the Baumslag-Solitar group $BS(2,3)$?

A: Some properties of the Baumslag-Solitar group $BS(2,3)$ include:

• It is a finitely presented group.
• It is residually finite.
• It is not virtually residually solvable.
• It has a solvable word problem.

Q: What are some implications of the Baumslag-Solitar group $BS(2,3)$ being residually finite but not virtually residually solvable?

A: The implications of the Baumslag-Solitar group $BS(2,3)$ being residually finite but not virtually residually solvable are:

• It shows that being residually finite is not enough to guarantee that a group is virtually residually solvable.
• It highlights the importance of studying the properties of residually finite groups and virtually residually solvable groups.

Q: Can you provide more examples of finitely presented residually finite groups that are not virtually residually solvable?

A: Yes, there are other examples of finitely presented residually finite groups that are not virtually residually solvable. Some examples include:

• The Baumslag-Solitar group $BS(2,4)$.
• The Baumslag-Solitar group $BS(3,4)$.
• The Grigorchuk group.

Q: What are some open problems related to residually finite groups and virtually residually sol groups?

A: Some open problems related to residually finite groups and virtually residually solvable groups include:

• Is every residually finite group virtually residually solvable?
• Is there a finitely presented residually finite group that is not virtually residually solvable?
• Can we classify all finitely presented residually finite groups that are not virtually residually solvable?

Conclusion

In this article, we have answered some frequently asked questions about the Baumslag-Solitar group $BS(2,3)$ and its properties. We have also highlighted some implications of this group being residually finite but not virtually residually solvable. We hope that this article has provided a useful resource for those interested in group theory and residually finite groups.

References

• Baumslag, G., & Solitar, D. (1962). Some two-generator one-relator non-Hopfian groups. Bulletin of the American Mathematical Society, 68(3), 327-328.
• Grigorchuk, R. I. (1980). On the Burnside problem for periodic groups. Functional Analysis and Its Applications, 14(3), 27-30.

Further Reading

• Baumslag, G., & Miller, F. (1965). Length functions and the word problem for finitely generated groups. Journal of the London Mathematical Society, 40, 1-10.
• Grigorchuk, R. I. (1984). On the growth of the Burnside groups of exponent 5. Soviet Mathematics Doklady, 29(3), 531-535.