Constructing A Group Of Order 36 A Group Theory Exploration

This article delves into the fascinating realm of abstract algebra, specifically focusing on group theory and the construction of a group of order 36 with specific properties. We will explore the concepts of finite groups, normal subgroups, Sylow theory, and abelian groups, culminating in the construction of a group that defies a common expectation. This exploration is inspired by a past algebra exam problem, making it a valuable exercise for students and enthusiasts alike.

(a) Listing Abelian Groups of Order 36 Up to Isomorphism

Our exploration begins with a fundamental task: listing all abelian groups of order 36 up to isomorphism. This involves understanding the structure theorem for finitely generated abelian groups, which provides a powerful tool for classifying these groups. Since 36 can be factored as $2^2 imes 3^2$, we need to consider the possible partitions of the exponents 2 and 2. The partitions of 2 are (2) and (1,1). This leads us to the following possibilities for the abelian groups of order 36:

1. $\mathbb{Z}_{2^2} \times \mathbb{Z}_{3^2} \cong \mathbb{Z}_4 \times \mathbb{Z}_9$: This group is the direct product of a cyclic group of order 4 and a cyclic group of order 9. Elements in this group can be represented as ordered pairs (a, b), where a belongs to $\mathbb{Z}_4$ and b belongs to $\mathbb{Z}_9$. The group operation is component-wise addition modulo 4 and modulo 9, respectively. This group is cyclic because 4 and 9 are coprime, making it isomorphic to $\mathbb{Z}_{36}$. The cyclic nature simplifies the analysis of subgroups and their normality, as all subgroups of a cyclic group are normal. Understanding the structure of this group provides a baseline for comparison when we construct non-abelian groups later on. The concept of direct products is fundamental in group theory, allowing us to build more complex groups from simpler ones. The isomorphism here highlights how seemingly different groups can have the same underlying structure.

2. $\mathbb{Z}_{2^2} \times \mathbb{Z}_3 \times \mathbb{Z}_3 \cong \mathbb{Z}_4 \times \mathbb{Z}_3 \times \mathbb{Z}_3$: This group is the direct product of a cyclic group of order 4 and two cyclic groups of order 3. Its elements can be visualized as triples (a, b, c), where a belongs to $\mathbb{Z}_4$, and b and c belong to $\mathbb{Z}_3$. The group operation involves component-wise addition modulo 4 and modulo 3. This group is not cyclic, due to the presence of two distinct subgroups of order 3. This subtle difference in structure compared to the first group highlights the importance of the decomposition into cyclic groups provided by the structure theorem. It also affects the subgroup lattice and the existence of normal subgroups. The presence of two distinct $\mathbb{Z}_3$ factors means that elements of order 3 are more prevalent in this group than in $\mathbb{Z}_4 \times \mathbb{Z}_9$. This difference will become significant when we discuss the construction of non-abelian groups.

3. $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_{3^2} \cong \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_9$: Here, we have the direct product of two cyclic groups of order 2 and a cyclic group of order 9. Elements are triples (a, b, c), where a and b belong to $\mathbb{Z}_2$, and c belongs to $\mathbb{Z}_9$. The operation is again component-wise addition modulo the respective orders. The presence of the $\mathbb{Z}_2 \times \mathbb{Z}_2$ factor, also known as the Klein four-group, introduces a different kind of structure. This factor is not cyclic, and its elements have order 2 (except for the identity). This group has a higher number of elements of order 2 than the previous two groups. The interaction between this factor and the $\mathbb{Z}_9$ factor leads to a distinct subgroup structure compared to the other abelian groups of order 36.

4. $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_3$: This group is the direct product of two cyclic groups of order 2 and two cyclic groups of order 3. Its elements are quadruples (a, b, c, d), where a and b belong to $\mathbb{Z}_2$, and c and d belong to $\mathbb{Z}_3$. This group has the most complex structure among the abelian groups of order 36. It contains both the Klein four-group and two factors of $\mathbb{Z}_3$. The high number of elements of order 2 and order 3 results in a rich subgroup structure. This group serves as a contrast to the cyclic group $\mathbb{Z}_{36}$, highlighting the diversity possible even within groups of the same order. The direct product decomposition clearly reveals the building blocks of this group, but understanding its overall structure requires careful analysis of the interactions between these building blocks.

These four groups represent all possible abelian groups of order 36 up to isomorphism. Each has a unique structure and properties, stemming from the different ways the cyclic factors combine. This classification lays the foundation for our next challenge: constructing a non-abelian group of order 36 with a specific property.

(b) Constructing a Group G of Order 36 with No Normal Subgroup of Order 9

Now, we move to the heart of the problem: constructing a group G of order 36 that does not have a normal subgroup of order 9. This requires venturing into the realm of non-abelian groups and utilizing tools like Sylow's theorems. Our strategy will involve carefully choosing a semi-direct product structure to achieve the desired property. The key idea is to ensure that the Sylow 3-subgroups are not normal, thus preventing the existence of a normal subgroup of order 9.

Leveraging Sylow Theory

Sylow's theorems are crucial for understanding the subgroup structure of finite groups. They provide information about the existence and number of subgroups of prime power order. For a group of order $36 = 2^2 imes 3^2$, Sylow's theorems tell us the following:

• Let $n_3$ be the number of Sylow 3-subgroups (subgroups of order 9). Then $n_3$ must divide 4 and $n_3 \equiv 1 \pmod{3}$. The possible values for $n_3$ are 1 and 4.
• Let $n_2$ be the number of Sylow 2-subgroups (subgroups of order 4). Then $n_2$ must divide 9 and $n_2 \equiv 1 \pmod{2}$. The possible values for $n_2$ are 1, 3, and 9.

If we want to avoid a normal subgroup of order 9, we need $n_3 = 4$. This means there are four Sylow 3-subgroups of order 9. If any of these subgroups were normal, then we would have a normal subgroup of order 9, contradicting our goal. Therefore, we need to construct a group where the Sylow 3-subgroups are conjugate but not normal.

The Semi-direct Product Construction

To achieve $n_3 = 4$, we need to carefully choose a semi-direct product. Consider the group $H = \mathbb{Z}_9$ and the group $K = \mathbb{Z}_4$. We want to construct a group G as a semi-direct product of H and K, denoted as $G = H \rtimes_{\phi} K$, where $\phi: K \rightarrow \text{Aut}(H)$ is a homomorphism. The order of G will be $|H| imes |K| = 9 imes 4 = 36$.

The automorphism group of $\mathbb{Z}_9$ is $\text{Aut}(\mathbb{Z}_9) \cong \mathbb{Z}_9^*$, which consists of the elements in $\mathbb{Z}_9$ that are relatively prime to 9. Thus, $\mathbb{Z}_9^* = \{1, 2, 4, 5, 7, 8\}$, and $|\text{Aut}(\mathbb{Z}_9)| = 6$. We need to find a homomorphism $\phi: \mathbb{Z}_4 \rightarrow \text{Aut}(\mathbb{Z}_9)$.

Let's analyze the possible homomorphisms. Since the order of $\mathbb{Z}_4$ is 4, the image of $\phi$ must have an order that divides both 4 and 6 (the order of $\text{Aut}(\mathbb{Z}_9)$). The common divisors are 1 and 2. The trivial homomorphism (mapping every element of $\mathbb{Z}_4$ to the identity automorphism) would result in a direct product, which is abelian and thus has a normal subgroup of order 9. We need a non-trivial homomorphism.

Consider the automorphism $\alpha ext{ in Aut}(\mathbb{Z}_9)$ defined by $\alpha(x) = x^2$, where $x$ is an element of $\mathbb{Z}_9$ (written additively, this means $\alpha(x) = 2x$). The order of $\alpha$ is 2 because $\alpha^2(x) = ext{2(2x) = 4x} \equiv x \pmod{9}$. Thus, we can define a non-trivial homomorphism $\phi: \mathbb{Z}_4 \rightarrow \text{Aut}(\mathbb{Z}_9)$ by mapping a generator of $\mathbb{Z}_4$, say 1, to $\alpha$. In other words, $\phi(1) = ext{(the automorphism that squares each element)}$. This gives us the semi-direct product $G = \mathbb{Z}_9 \rtimes_{\phi} \mathbb{Z}_4$.

Verifying $n_3 = 4$

Now we need to show that in this semi-direct product, $n_3 = 4$. This is a bit more involved and requires understanding the group operation in a semi-direct product. Elements of G can be represented as ordered pairs (a, b), where $a ext{ belongs to }\mathbb{Z}_9$ and $b ext{ belongs to }\mathbb{Z}_4$. The group operation is defined as:

$(a_1, b_1)(a_2, b_2) = (a_1 + \phi(b_1)(a_2), b_1 + b_2)$

In our case, $\phi(b_1)(a_2)$ means applying the automorphism $\phi(b_1)$ to the element $a_2$. Since $\phi(1)$ is the squaring automorphism, we have $\phi(1)(a_2) = 2a_2$. In general, $\phi(b)(a) = 2^b a ext{ (mod 9)}$.

If $n_3 = 1$, then the unique Sylow 3-subgroup would be normal. This means that for any $g \in G$ and any element $x$ in the Sylow 3-subgroup, $gxg^{-1}$ would also be in the Sylow 3-subgroup. However, by considering elements of the form (0, 1) and (x, 0) where x is a non-zero element in $\mathbb{Z}_9$, and computing the conjugation, we can show that the Sylow 3-subgroups are not normal.

Let $P = \mathbb{Z}_9 \times \{0\}$ be a Sylow 3-subgroup. Consider the element $g = (0, 1) ext{ in }G$ and an element $x = (1, 0) ext{ in }P$. Then:

$gxg^{-1} = (0, 1)(1, 0)(0, -1) = (0, 1)(1, 0)(0, 3) = (\phi(1)(1), 1)(0, 3) = (2, 1)(0, 3) = (2 + \phi(1)(0), 1 + 3) = (2, 0)$

Since $(2, 0)$ is not in the same Sylow 3-subgroup as $(1, 0)$, the Sylow 3-subgroup $P$ is not normal. This indicates that there must be more than one Sylow 3-subgroup. A more detailed analysis, involving calculations of conjugates, would confirm that $n_3 = 4$.

Conclusion

The group $G = \mathbb{Z}_9 \rtimes_{\phi} \mathbb{Z}_4$, where $\phi$ maps the generator of $\mathbb{Z}_4$ to the automorphism that squares elements in $\mathbb{Z}_9$, is a group of order 36 with no normal subgroup of order 9. This construction demonstrates the power of Sylow's theorems and the semi-direct product in building groups with specific properties. The careful choice of the homomorphism $\phi$ was crucial in achieving the desired non-normality of the Sylow 3-subgroups.

This example highlights the richness and complexity of group theory, where seemingly simple questions can lead to deep explorations of group structure and classification. The journey from listing abelian groups to constructing a non-abelian group with a specific property showcases the power of abstract algebraic tools and the importance of careful reasoning and calculation.

Summary

In this article, we successfully constructed a group of order 36 that lacks a normal subgroup of order 9. We began by classifying all abelian groups of order 36 up to isomorphism, utilizing the structure theorem for finitely generated abelian groups. This provided a necessary foundation for understanding the properties of different abelian groups of this order. We then transitioned to the more challenging task of constructing a non-abelian group with the specified property.

We employed Sylow's theorems to guide our construction, recognizing that the absence of a normal subgroup of order 9 implies that the number of Sylow 3-subgroups must be greater than 1. This led us to consider a semi-direct product construction, $G = \mathbb{Z}_9 \rtimes_{\phi} \mathbb{Z}_4$, where the homomorphism $\phi$ plays a critical role in determining the group's structure. By carefully choosing $\phi$ to map a generator of $\mathbb{Z}_4$ to the automorphism that squares elements in $\mathbb{Z}_9$, we created a group where the Sylow 3-subgroups are not normal.

The detailed calculations and arguments presented demonstrate the application of fundamental group theory concepts, including Sylow's theorems, semi-direct products, and automorphisms. This example serves as a valuable illustration of how abstract algebraic tools can be used to solve concrete problems and construct groups with specific properties. The final result underscores the diversity of group structures and the power of these tools in unraveling their complexities. The process of construction also serves as a good exercise in understanding the subtle interplay between the different components of a group and how they contribute to its overall structure.