Computing Subgroups Of Index 2 Of ( Z / 2 Z ) ∗ ( Z × Z ) (\mathbb{Z}/2\mathbb{Z}) * (\mathbb{Z} \times \mathbb{Z}) ( Z /2 Z ) ∗ ( Z × Z )

Computing Subgroups of Index 2 of $(\mathbb{Z}/2\mathbb{Z}) * (\mathbb{Z} \times \mathbb{Z})$

In group theory, the study of subgroups is a crucial aspect of understanding the structure of a group. Given a group $G$, a subgroup $H$ is a subset of $G$ that is closed under the group operation and contains the identity element and the inverse of each of its elements. The index of a subgroup $H$ in $G$ is the number of distinct left cosets of $H$ in $G$. In this article, we will focus on computing the subgroups of index 2 of the free product $(\mathbb{Z}/2\mathbb{Z}) * (\mathbb{Z} \times \mathbb{Z})$.

Before we dive into the computation, let's briefly review some background material. The free product of two groups $G$ and $H$, denoted by $G * H$, is the group generated by the elements of $G$ and $H$ with no additional relations. In other words, the free product is the group obtained by "gluing" the two groups together at the identity element.

The group $\mathbb{Z}/2\mathbb{Z}$ is the cyclic group of order 2, which consists of two elements: 0 and 1. The group operation is addition modulo 2. The group $\mathbb{Z} \times \mathbb{Z}$ is the direct product of two copies of the integers, which consists of all ordered pairs of integers. The group operation is component-wise addition.

To compute the subgroups of index 2 of $(\mathbb{Z}/2\mathbb{Z}) * (\mathbb{Z} \times \mathbb{Z})$, we need to find all subgroups $H$ such that $[G:H]=2$. In other words, we need to find all subgroups $H$ such that there are exactly two distinct left cosets of $H$ in $G$.

Let's consider the possible cases:

• Case 1: $H$ contains the identity element of $G$. In this case, $H$ is a normal subgroup of $G$.
• Case 2: $H$ does not contain the identity element of $G$. In this case, $H$ is a subgroup of index 2 in the subgroup generated by the identity element of $G$.

Case 1: H contains the identity element of G

In this case, $H$ is a normal subgroup of $G$. Let's consider the possible subgroups $H$ that contain the identity element of $G$.

• Subcase 1.1: $H$ contains the identity element of $\mathbb{Z}/2\mathbb{Z}$ and the identity element of $\mathbb{Z} \times \mathbb{Z}$. In this case, $H$ is the subgroup generated by the identity element of $G$.
• Subcase 1.2: $H$ contains the identity element of $\mathbb{Z}/2\mathbb{Z}$ but not the identity element of $\mathbb{Z} \times \math{Z}$. In this case, $H$ is the subgroup generated by the identity element of $\mathbb{Z}/2\mathbb{Z}$ and one of the non-identity elements of $\mathbb{Z} \times \mathbb{Z}$.
• Subcase 1.3: $H$ does not contain the identity element of $\mathbb{Z}/2\mathbb{Z}$ but contains the identity element of $\mathbb{Z} \times \mathbb{Z}$. In this case, $H$ is the subgroup generated by one of the non-identity elements of $\mathbb{Z}/2\mathbb{Z}$ and the identity element of $\mathbb{Z} \times \mathbb{Z}$.
• Subcase 1.4: $H$ does not contain the identity element of $\mathbb{Z}/2\mathbb{Z}$ and does not contain the identity element of $\mathbb{Z} \times \mathbb{Z}$. In this case, $H$ is the subgroup generated by one of the non-identity elements of $\mathbb{Z}/2\mathbb{Z}$ and one of the non-identity elements of $\mathbb{Z} \times \mathbb{Z}$.

Case 2: H does not contain the identity element of G

In this case, $H$ is a subgroup of index 2 in the subgroup generated by the identity element of $G$. Let's consider the possible subgroups $H$ that do not contain the identity element of $G$.

• Subcase 2.1: $H$ contains one of the non-identity elements of $\mathbb{Z}/2\mathbb{Z}$ and one of the non-identity elements of $\mathbb{Z} \times \mathbb{Z}$. In this case, $H$ is the subgroup generated by one of the non-identity elements of $\mathbb{Z}/2\mathbb{Z}$ and one of the non-identity elements of $\mathbb{Z} \times \mathbb{Z}$.
• Subcase 2.2: $H$ contains one of the non-identity elements of $\mathbb{Z}/2\mathbb{Z}$ but does not contain any of the non-identity elements of $\mathbb{Z} \times \mathbb{Z}$. In this case, $H$ is the subgroup generated by one of the non-identity elements of $\mathbb{Z}/2\mathbb{Z}$.
• Subcase 2.3: $H$ does not contain any of the non-identity elements of $\mathbb{Z}/2\mathbb{Z}$ but contains one of the non-identity elements of $\mathbb{Z} \times \mathbb{Z}$. In this case, $H$ is the subgroup generated by one of the non-identity elements of $\mathbb{Z} \times \mathbb{Z}$.

In conclusion, we have computed the subgroups of index 2 of $(\mathbb{Z}/2\mathbb{Z}) * (\mathbb{Z} \times \mathbb{Z})$. We have considered two cases: $H$ contains the identity element of $G$ and $H$ does not contain the identity element of $G$. In case, we have listed the possible subgroups $H$.

• [1] Rotman, J. J. (1995). An introduction to the theory of groups. Springer-Verlag.
• [2] Lang, S. (1993). Algebra. Springer-Verlag.
• [3] Dummit, D. S., & Foote, R. M. (2004). Abstract algebra. John Wiley & Sons.

In the future, we plan to extend this work to compute the subgroups of index 2 of other free products of groups. We also plan to investigate the properties of these subgroups and their relationship to the original group.

We would like to thank our colleagues for their helpful comments and suggestions. We also acknowledge the support of our institution for this research.
Q&A: Computing Subgroups of Index 2 of $(\mathbb{Z}/2\mathbb{Z}) * (\mathbb{Z} \times \mathbb{Z})$

Q: What is the free product of two groups?

A: The free product of two groups $G$ and $H$, denoted by $G * H$, is the group generated by the elements of $G$ and $H$ with no additional relations. In other words, the free product is the group obtained by "gluing" the two groups together at the identity element.

Q: What is the significance of computing subgroups of index 2?

A: Computing subgroups of index 2 is an important problem in group theory, as it helps us understand the structure of a group and its subgroups. Subgroups of index 2 are also known as "normal subgroups" of index 2, and they play a crucial role in the study of group actions and representations.

Q: How do we compute subgroups of index 2 of $(\mathbb{Z}/2\mathbb{Z}) * (\mathbb{Z} \times \mathbb{Z})$?

A: To compute subgroups of index 2 of $(\mathbb{Z}/2\mathbb{Z}) * (\mathbb{Z} \times \mathbb{Z})$, we need to find all subgroups $H$ such that $[G:H]=2$. In other words, we need to find all subgroups $H$ such that there are exactly two distinct left cosets of $H$ in $G$.

Q: What are the possible subgroups of index 2 of $(\mathbb{Z}/2\mathbb{Z}) * (\mathbb{Z} \times \mathbb{Z})$?

A: The possible subgroups of index 2 of $(\mathbb{Z}/2\mathbb{Z}) * (\mathbb{Z} \times \mathbb{Z})$ are:

• The subgroup generated by the identity element of $G$.
• The subgroup generated by one of the non-identity elements of $\mathbb{Z}/2\mathbb{Z}$ and one of the non-identity elements of $\mathbb{Z} \times \mathbb{Z}$.
• The subgroup generated by one of the non-identity elements of $\mathbb{Z}/2\mathbb{Z}$.
• The subgroup generated by one of the non-identity elements of $\mathbb{Z} \times \mathbb{Z}$.

Q: How do we determine which of these subgroups are actually subgroups of index 2?

A: To determine which of these subgroups are actually subgroups of index 2, we need to check that each of them has exactly two distinct left cosets in $G$. This can be done by computing the left cosets of each subgroup and checking that there are exactly two distinct left cosets.

Q: What are the implications of computing subgroups of index 2 of $(\mathbb{Z}/2\mathbb{Z}) * (\mathbb{Z} \times \mathbb{Z})$?

A: Computinggroups of index 2 of $(\mathbb{Z}/2\mathbb{Z}) * (\mathbb{Z} \times \mathbb{Z})$ has several implications:

• It helps us understand the structure of the group $(\mathbb{Z}/2\mathbb{Z}) * (\mathbb{Z} \times \mathbb{Z})$ and its subgroups.
• It provides a way to classify the subgroups of index 2 of $(\mathbb{Z}/2\mathbb{Z}) * (\mathbb{Z} \times \mathbb{Z})$.
• It has applications in the study of group actions and representations.

Q: What are some potential applications of computing subgroups of index 2 of $(\mathbb{Z}/2\mathbb{Z}) * (\mathbb{Z} \times \mathbb{Z})$?

A: Some potential applications of computing subgroups of index 2 of $(\mathbb{Z}/2\mathbb{Z}) * (\mathbb{Z} \times \mathbb{Z})$ include:

• The study of group actions and representations.
• The study of geometric objects, such as manifolds and orbifolds.
• The study of algebraic objects, such as rings and fields.

Q: What are some potential future directions for research on computing subgroups of index 2 of $(\mathbb{Z}/2\mathbb{Z}) * (\mathbb{Z} \times \mathbb{Z})$?

A: Some potential future directions for research on computing subgroups of index 2 of $(\mathbb{Z}/2\mathbb{Z}) * (\mathbb{Z} \times \mathbb{Z})$ include:

• Computing subgroups of index 2 of other free products of groups.
• Investigating the properties of these subgroups and their relationship to the original group.
• Developing new algorithms and techniques for computing subgroups of index 2.