Character Table Of Semidirect Product Of Z / 7 Z \mathbb{Z}/7\mathbb{Z} Z /7 Z With Z / 3 Z \mathbb{Z}/3\mathbb{Z} Z /3 Z

Introduction to Semidirect Products and Character Tables

In the realm of group theory, particularly in the study of finite groups, the semidirect product stands as a crucial construction that enriches our understanding of group structures. When diving into the representation theory of these groups, we often encounter the challenge of computing their character tables. Character tables are vital tools that summarize the irreducible representations of a group, providing insights into its structure and properties. In this article, we delve deep into computing the character table of a specific semidirect product: $\mathbb{Z}/7\mathbb{Z} \rtimes \mathbb{Z}/3\mathbb{Z}$. This group is defined by the relation $bab^{-1} = a^2$, where $a$ is a generator of $\mathbb{Z}/7\mathbb{Z}$ and $b$ is a generator of $\mathbb{Z}/3\mathbb{Z}$.

The significance of understanding and computing character tables extends to various areas of mathematics and physics. In mathematics, character tables help in classifying groups and understanding their automorphism groups. In physics, they play a pivotal role in quantum mechanics, particularly in analyzing the symmetry properties of physical systems. For instance, in molecular physics, character tables are used to determine the vibrational modes of molecules. The process of constructing a character table involves several steps, each requiring a solid grasp of group theory fundamentals and representation theory. This includes determining the conjugacy classes of the group, finding the irreducible representations, and computing the characters of these representations. The intricacies of this process highlight the importance of a systematic approach, especially for non-abelian groups where the representations are not necessarily one-dimensional.

The semidirect product $\mathbb{Z}/7\mathbb{Z} \rtimes \mathbb{Z}/3\mathbb{Z}$, denoted as $G$, presents an interesting case study due to its non-abelian nature, which arises from the non-trivial action of $\mathbb{Z}/3\mathbb{Z}$ on $\mathbb{Z}/7\mathbb{Z}$. The relation $bab^{-1} = a^2$ dictates how the elements of $\mathbb{Z}/3\mathbb{Z}$ transform the elements of $\mathbb{Z}/7\mathbb{Z}$, leading to a more complex group structure than a direct product. This complexity necessitates a careful examination of the group's structure to determine its conjugacy classes and irreducible representations. Our journey to compute the character table of $G$ will involve several key steps, starting with identifying the group's structure, then determining its conjugacy classes, and finally, constructing the irreducible representations and their characters. This detailed exploration not only provides a concrete example of character table computation but also reinforces the fundamental concepts of group theory and representation theory.

Group Structure and Presentation

To embark on the computation of the character table for the semidirect product $G = \mathbb{Z}/7\mathbb{Z} \rtimes \mathbb{Z}/3\mathbb{Z}$, a firm grasp of the group's structure is paramount. The group $G$ is defined by the presentation $\langle a, b \mid a^7 = 1, b^3 = 1, bab^{-1} = a^2 \rangle$. Here, $a$ generates the cyclic group of order 7, $\mathbb{Z}/7\mathbb{Z}$, and $b$ generates the cyclic group of order 3, $\mathbb{Z}/3\mathbb{Z}$. The relation $bab^{-1} = a^2$ signifies the action of $\mathbb{Z}/3\mathbb{Z}$ on $\mathbb{Z}/7\mathbb{Z}$, making the product semidirect rather than direct. This action is crucial as it determines how the two subgroups interact within the larger group $G$. Understanding this interaction is the first step in deciphering the group's overall structure and representation theory.

The order of the group $G$ is the product of the orders of $\mathbb{Z}/7\mathbb{Z}$ and $\mathbb{Z}/3\mathbb{Z}$, which is $|G| = 7 \times 3 = 21$. This finite order allows us to methodically explore the group's elements and their relationships. The general element of $G$ can be written in the form $a^i b^j$, where $0 \leq i < 7$ and $0 \leq j < 3$. The relation $bab^{-1} = a^2$ is the key to simplifying products of elements in $G$. By repeatedly applying this relation, we can express any element in the form $a^i b^j$. For example, to compute the product $(a^i b^j)(a^k b^l)$, we use the relation to move $b^j$ past $a^k$, which results in $a^i b^j a^k b^l = a^i (b^j a^k b^{-j}) b^{j+l}$. The term $b^j a^k b^{-j}$ can be simplified using the given relation multiple times, illustrating the intricate interplay between the elements of $\mathbb{Z}/7\mathbb{Z}$ and $\mathbb{Z}/3\mathbb{Z}$.

The non-commutativity of $G$ is a direct consequence of the relation $bab^{-1} = a^2$. If $G$ were abelian, we would have $bab^{-1} = a$, but the relation specifies $a^2$, indicating a non-trivial action and thus a non-abelian group structure. This non-commutativity has significant implications for the character table, as it means that not all irreducible representations will be one-dimensional. The dimensions of the irreducible representations and their characters will provide a deeper understanding of the group's structure. Before constructing the character table, we must first identify the conjugacy classes of $G$. These classes partition the group into sets of elements that are conjugate to each other, and they form the foundation upon which the character table is built. The next section will delve into the process of determining the conjugacy classes of $G$, paving the way for the subsequent computation of its character table.

Conjugacy Classes of G

Determining the conjugacy classes of $G = \mathbb{Z}/7\mathbb{Z} \rtimes \mathbb{Z}/3\mathbb{Z}$ is a critical step in constructing its character table. Conjugacy classes partition a group into subsets of elements that are conjugate to each other. Two elements, $g$ and $h$, in a group are conjugate if there exists an element $x$ in the group such that $g = xhx^{-1}$. The conjugacy class of an element $h$ is the set of all elements conjugate to $h$, denoted as $Cl(h) = \{ xhx^{-1} \mid x \in G \}$. Understanding these classes is vital because elements within the same class have the same character values for any given representation. This property significantly simplifies the process of computing character tables.

To find the conjugacy classes of $G$, we examine how elements of the form $a^i b^j$ conjugate other elements. Recall that the general element of $G$ is of the form $a^i b^j$ where $0 \leq i < 7$ and $0 \leq j < 3$, and the defining relation is $bab^{-1} = a^2$. We start by considering the conjugacy class of the identity element, $e$, which trivially contains only the identity itself, i.e., $Cl(e) = \{ e \}$. Next, we investigate the conjugacy class of $a$. To do this, we consider conjugating $a$ by the generators $a$ and $b$. Conjugating $a$ by $a$ gives $a a a^{-1} = a$, which is expected. Conjugating $a$ by $b$ gives $bab^{-1} = a^2$, indicating that $a$ and $a^2$ are in the same conjugacy class. Continuing this process, we find:

• $b a^2 b^{-1} = (bab^{-1})^2 = (a^2)^2 = a^4$
• $b a^4 b^{-1} = (bab^{-1})^4 = (a^2)^4 = a^8 = a$ (since $a^7 = 1$)

Thus, the elements $a, a^2, a^4$ form a conjugacy class. Similarly, we can find the conjugacy class of $a^3$:

• $b a^3 b^{-1} = (bab^{-1})^3 = (a^2)^3 = a^6$
• $b a^6 b^{-1} = (bab^{-1})^6 = (a^2)^6 = a^{12} = a^5$
• $b a^5 b^{-1} = (bab^{-1})^5 = (a^2)^5 = a^{10} = a^3$

This shows that $a^3, a^6, a^5$ form another conjugacy class. Now we consider the conjugacy class of $b$. Conjugating $b$ by $a$ yields $aba^{-1}$. To simplify this, we use the relation $bab^{-1} = a^2$, which implies $b = a^{-1}ba^2$. Rearranging, we get $a^{-1}b = ba^{-2}$, and multiplying by $a$ on the left gives $b = aba^{-2}$, so $aba^{-1} = ba$. Conjugating $ba$ by $a$ gives $a(ba)a^{-1} = a^2ba^{-1}$. Continuing this process can be complex, so it's more efficient to consider conjugating $b$ directly. Conjugating $b$ by $a^i$ gives $a^i b a^{-i}$. Using the relation $bab^{-1} = a^2$, we can derive $a^i b = b a^{2^i}$. Thus, $a^i b a^{-i} = b a^{2^i} a^{-i} = b a^{2^i - i}$.

For $i = 1$, we have $aba^{-1} = ba$. For $i = 2$, we have $a^2 b a^{-2} = b a^{2^2 - 2} = b a^2$. For $i = 3$, we have $a^3 b a^{-3} = b a^{2^3 - 3} = b a^5$. This suggests that the conjugacy class of $b$ might involve terms of the form $ba^k$. To confirm, we consider conjugating $b$ by $b$ itself, which trivially gives $b$. Thus, the conjugacy class of $b$ includes $b$, $ba$, and $ba^2$. The same logic applies to $b^2$, and we can similarly find its conjugacy class. In summary, the conjugacy classes of $G$ are:

1. $\Cl(e) = \{ e \}$
2. $\Cl(a) = \{ a, a^2, a^4 \}$
3. $\Cl(a^3) = \{ a^3, a^5, a^6 \}$
4. $\Cl(b) = \{ b, a b, a^2 b, a^3 b, a^4 b, a^5 b, a^6 b \}$
5. $\Cl(b^2) = \{ b^2, a b^2, a^2 b^2, a^3 b^2, a^4 b^2, a^5 b^2, a^6 b^2 \}$

Counting the elements in each class, we have 1 + 3 + 3 + 7 + 7 = 21, which matches the order of the group, confirming that we have identified all conjugacy classes correctly. The number of conjugacy classes, which is 5 in this case, is equal to the number of irreducible representations of the group. This is a fundamental result in representation theory. With the conjugacy classes determined, we can now proceed to find the irreducible representations and their characters, ultimately constructing the character table of $G$.

Irreducible Representations and Characters

With the conjugacy classes of $G = \mathbb{Z}/7\mathbb{Z} \rtimes \mathbb{Z}/3\mathbb{Z}$ identified, the next crucial step is to determine the irreducible representations and their corresponding characters. The character of a representation is a function that maps each group element to the trace of the matrix representing that element in the representation. Irreducible representations are representations that cannot be decomposed into smaller representations, and they form the building blocks of all representations of a group. The character table, which tabulates the characters of all irreducible representations for each conjugacy class, provides a comprehensive summary of the group's representation-theoretic properties. Since $G$ has 5 conjugacy classes, it has 5 irreducible representations.

The dimensions of the irreducible representations, denoted as $d_i$, must satisfy the equation $\sum_{i=1}^{5} d_i^2 = |G| = 21$. Furthermore, the number of one-dimensional representations is equal to the index of the commutator subgroup $G'$ in $G$, i.e., the order of the abelianization $G/G'$. To find the commutator subgroup $G'$, we consider the commutators of the generators $a$ and $b$. The commutator of $a$ and $b$ is $[a, b] = aba^{-1}b^{-1}$. Using the relation $bab^{-1} = a^2$, we have $b^{-1}a^{-1} = a^{-2}b^{-1}$. Thus, $[a, b] = a(ba^{-1}b^{-1}) = aa^{-2} = a^{-1}$, implying that the subgroup generated by $a$ is contained in the commutator subgroup. Since the subgroup generated by $a$ has order 7, and it is a normal subgroup, it is likely to be the commutator subgroup itself. The quotient group $G/\langle a \rangle$ is isomorphic to $\mathbb{Z}/3\mathbb{Z}$, which has order 3. Therefore, the order of $G'$ is 7, and the abelianization $G/G'$ has order 3, indicating that there are 3 one-dimensional representations.

The one-dimensional representations are homomorphisms from $G$ to $\mathbb{C}^*$, the multiplicative group of nonzero complex numbers. Since $G/G' \cong \mathbb{Z}/3\mathbb{Z}$, the one-dimensional representations correspond to the representations of $\mathbb{Z}/3\mathbb{Z}$. Let $\omega = e^{2\pi i/3}$ be a primitive cube root of unity. The characters of the three one-dimensional representations, $\chi_1$, $\chi_2$, and $\chi_3$, can be defined as follows:

• $\chi_1(a^i b^j) = 1$ for all $i, j$ (the trivial representation)
• $\chi_2(a^i b^j) = \omega^j$
• $\chi_3(a^i b^j) = \omega^{2j}$

These characters are constant on the conjugacy classes, and their values can be easily determined for each class. Now, we need to find the two remaining irreducible representations. Let their dimensions be $d_4$ and $d_5$. From the equation $\sum_{i=1}^{5} d_i^2 = 21$, we have $1^2 + 1^2 + 1^2 + d_4^2 + d_5^2 = 21$, which simplifies to $d_4^2 + d_5^2 = 18$. The only positive integer solutions for this equation are $d_4 = d_5 = 3$. Thus, we have two 3-dimensional irreducible representations.

To construct these 3-dimensional representations, we look for subgroups of $G$ that can induce up to a 3-dimensional representation. Consider the normal subgroup $\mathbb{Z}/7\mathbb{Z}$, generated by $a$. The irreducible representations of $\mathbb{Z}/7\mathbb{Z}$ are one-dimensional, and their characters are given by $\psi_k(a) = e^{2\pi i k/7}$, where $k = 0, 1, ..., 6$. We can induce these representations to $G$. Let $\psi_1$ be the representation of $\langle a \rangle$ with character $\psi_1(a) = e^{2\pi i/7}$. The induced representation from $\psi_1$ to $G$ has dimension $[G : \langle a \rangle] = 3$. This induced representation is irreducible. Similarly, we can induce the representation with character $\psi_3(a) = e^{6\pi i/7}$, which also yields a 3-dimensional irreducible representation.

The characters of these 3-dimensional representations, $\chi_4$ and $\chi_5$, can be computed using the induction formula. The character of the induced representation $\chi_4$ on the conjugacy class of $a$ is 0, since the elements in the conjugacy class of $a$ are not conjugate to $a$ within the subgroup $\langle a \rangle$. On the conjugacy class of $b$, the character is also 0. The character on the identity element is the dimension of the representation, which is 3. A similar process yields the characters for $\chi_5$. By carefully applying the induction formula and orthogonality relations for characters, we can complete the character table.

Construction of the Character Table

After determining the conjugacy classes and irreducible representations of $G = \mathbb{Z}/7\mathbb{Z} \rtimes \mathbb{Z}/3\mathbb{Z}$, we can now construct the character table. This table is a square matrix whose rows correspond to the irreducible representations and whose columns correspond to the conjugacy classes. The entry in the table at the intersection of a row and a column is the character value of the corresponding representation for an element in the corresponding conjugacy class. The character table encapsulates a wealth of information about the group and its representations, making it a powerful tool for further analysis.

Recall that we have identified five conjugacy classes:

1. $\Cl(e) = \{ e \}$
2. $\Cl(a) = \{ a, a^2, a^4 \}$ (size 3)
3. $\Cl(a^3) = \{ a^3, a^5, a^6 \}$ (size 3)
4. $\Cl(b) = \{ b, ab, a^2b, a^3b, a^4b, a^5b, a^6b \}$ (size 7)
5. $\Cl(b^2) = \{ b^2, ab^2, a^2b^2, a^3b^2, a^4b^2, a^5b^2, a^6b^2 \}$ (size 7)

We also found three one-dimensional representations, $\chi_1$, $\chi_2$, and $\chi_3$, and two 3-dimensional representations, $\chi_4$ and $\chi_5$. The characters of the one-dimensional representations are:

• $\chi_1(a^i b^j) = 1$
• $\chi_2(a^i b^j) = \omega^j$
• $\chi_3(a^i b^j) = \omega^{2j}$, where $\omega = e^{2\pi i/3}$

To compute the characters of the 3-dimensional representations, we use the induction formula and the characters of the representations of the subgroup $\mathbb{Z}/7\mathbb{Z}$. Let $\psi_k(a) = e^{2\pi i k/7}$ for $k = 1, 2, ..., 6$ be the characters of the irreducible representations of $\mathbb{Z}/7\mathbb{Z}$. We induce the representations corresponding to $\psi_1$ and $\psi_3$ to obtain $\chi_4$ and $\chi_5$, respectively. The induced character $\chi_4$ is given by:

$$\chi_4(g) = \frac{1}{|\langle a \rangle|} \sum_{x \in G, xgx^{-1} \in \langle a \rangle} \psi_1(xgx^{-1})$$

Similarly, the induced character $\chi_5$ is given by:

$$\chi_5(g) = \frac{1}{|\langle a \rangle|} \sum_{x \in G, xgx^{-1} \in \langle a \rangle} \psi_3(xgx^{-1})$$

Applying these formulas, we find the characters of $\chi_4$ and $\chi_5$ for each conjugacy class. For example, for the conjugacy class of $a$, the characters $\chi_4(a)$ and $\chi_5(a)$ are obtained by summing over the conjugates of $a$ that lie in $\langle a \rangle$. After careful computation, we can fill in the character table:

Conjugacy Class	$\{e\}$	$\{a, a^2, a^4\}$	$\{a^3, a^5, a^6\}$	$\{b, ab, ..., a^6b\}$	$\{b^2, ab^2, ..., a^6b^2\}$
$\chi_1$	1	1	1	1	1
$\chi_2$	1	1	1	$\omega$	$\omega^2$
$\chi_3$	1	1	1	$\omega^2$	$\omega$
$\chi_4$	3	$\zeta + \zeta^2 + \zeta^4$	$\zeta^3 + \zeta^5 + \zeta^6$	0	0
$\chi_5$	3	$\zeta^3 + \zeta^5 + \zeta^6$	$\zeta + \zeta^2 + \zeta^4$	0	0

Where $\zeta = e^{2\pi i/7}$ is a primitive 7th root of unity.

This table provides a complete picture of the character theory of $G$. The rows are orthogonal, and the columns satisfy orthogonality relations weighted by the size of the conjugacy class. This character table can be used to decompose representations, compute tensor products, and analyze the structure of $G$ further. The construction of this character table for the semidirect product $\mathbb{Z}/7\mathbb{Z} \rtimes \mathbb{Z}/3\mathbb{Z}$ exemplifies the intricate interplay between group theory and representation theory, providing valuable insights into the nature of finite groups and their representations.

Applications and Significance of Character Tables

Character tables, like the one we constructed for $G = \mathbb{Z}/7\mathbb{Z} \rtimes \mathbb{Z}/3\mathbb{Z}$, are not merely abstract mathematical constructs; they are powerful tools with wide-ranging applications across various fields. In representation theory, character tables serve as a concise summary of a group's irreducible representations, providing a complete inventory of the fundamental ways in which a group can act linearly on vector spaces. This information is invaluable for understanding the structure of the group itself and its relationships with other groups. The entries in the character table, which are complex numbers, encode crucial information about the group's symmetry properties and its internal structure.

One of the primary applications of character tables lies in the decomposition of representations. Any representation of a finite group can be decomposed into a direct sum of irreducible representations. The character table provides the necessary information to determine the multiplicity of each irreducible representation in this decomposition. By computing the inner product of the character of the given representation with the characters of the irreducible representations, we can readily find the number of times each irreducible representation appears in the decomposition. This process is fundamental in many areas, including physics and chemistry, where representations describe the symmetries of physical systems.

In physics, character tables play a crucial role in quantum mechanics, particularly in the study of molecular symmetries and crystal structures. The symmetry group of a molecule, for instance, determines the selection rules for spectroscopic transitions. The character table of the symmetry group allows physicists to classify the vibrational modes of the molecule and predict which transitions are allowed or forbidden based on symmetry considerations. Similarly, in solid-state physics, character tables are used to analyze the electronic band structure of crystals, where the symmetry of the crystal lattice dictates the allowed energy levels of the electrons.

Furthermore, character tables are instrumental in understanding group extensions and cohomology theory. The character table can help determine whether a group can be written as an extension of one subgroup by another, and it provides information about the possible extensions. In cohomology theory, which studies the algebraic invariants of topological spaces, character tables are used to compute group cohomology, which has applications in various areas of mathematics and physics. The orthogonality relations of characters, which are fundamental properties encoded in the character table, are crucial for these computations.

Beyond these specific applications, character tables have a broader significance in the study of finite groups. They provide a complete invariant for distinguishing between linear representations of a group. Two representations with the same character are equivalent, meaning they are essentially the same representation expressed in different bases. This uniqueness property makes the character table a powerful tool for classifying and comparing representations. In algebraic number theory, character tables are used to study the Galois groups of field extensions, which are finite groups that describe the symmetries of the roots of polynomial equations. The character table can help in understanding the structure of these Galois groups and their connections to the arithmetic properties of the field extensions.

The construction of a character table, as we demonstrated for $G = \mathbb{Z}/7\mathbb{Z} \rtimes \mathbb{Z}/3\mathbb{Z}$, is a testament to the beauty and power of group theory. It requires a deep understanding of the group's structure, its conjugacy classes, and its irreducible representations. The resulting table is a compact and elegant summary of the group's representation-theoretic properties, with applications spanning diverse areas of science and mathematics. The character table serves as a bridge between abstract algebraic concepts and concrete physical phenomena, making it an indispensable tool for researchers and students alike.

Conclusion

In this comprehensive exploration, we have successfully constructed the character table for the semidirect product $G = \mathbb{Z}/7\mathbb{Z} \rtimes \mathbb{Z}/3\mathbb{Z}$, a quintessential example of a non-abelian finite group. Our journey began with a thorough examination of the group's structure, guided by its defining presentation $\langle a, b \mid a^7 = 1, b^3 = 1, bab^{-1} = a^2 \rangle$. Understanding this structure was paramount for the subsequent steps in our computation. We meticulously identified the conjugacy classes of $G$, which partition the group into sets of elements that are conjugate to each other. This step is crucial because elements within the same conjugacy class share the same character values, significantly simplifying the construction of the character table.

Next, we delved into the realm of representation theory, where we sought to determine the irreducible representations of $G$. We leveraged the fact that the number of irreducible representations is equal to the number of conjugacy classes, and we utilized the fundamental result that the sum of the squares of the dimensions of the irreducible representations equals the order of the group. This allowed us to deduce the dimensions of the irreducible representations and systematically construct them. We found three one-dimensional representations, which correspond to the representations of the abelianization of $G$, and two 3-dimensional representations, which we obtained by inducing representations from the normal subgroup $\mathbb{Z}/7\mathbb{Z}$.

With the conjugacy classes and irreducible representations in hand, we meticulously computed the characters, which are the traces of the matrices representing the group elements in the representations. We employed the induction formula and the orthogonality relations for characters to ensure the accuracy of our computations. The resulting character table, a square matrix with rows corresponding to the irreducible representations and columns corresponding to the conjugacy classes, provides a comprehensive summary of the representation-theoretic properties of $G$. This table is not just a collection of numbers; it is a powerful tool that encodes deep insights into the structure and symmetries of the group.

The character table of $G$ has a myriad of applications, as we have discussed. It can be used to decompose representations, compute tensor products, and analyze group extensions. In physics, it finds applications in quantum mechanics, particularly in the study of molecular symmetries and crystal structures. In mathematics, it is instrumental in algebraic number theory and cohomology theory. The construction and analysis of character tables are therefore fundamental skills for anyone working in these areas.

This exploration of the character table of $\mathbb{Z}/7\mathbb{Z} \rtimes \mathbb{Z}/3\mathbb{Z}$ serves as a testament to the beauty and utility of group theory. It showcases the power of abstract algebraic concepts in solving concrete problems and provides a valuable case study for understanding the representation theory of finite groups. The process of constructing the character table requires a deep understanding of group structure, representation theory fundamentals, and computational techniques. This endeavor not only enhances our knowledge of the specific group $G$ but also reinforces our grasp of the broader mathematical principles that underpin the field of group theory.