Primitive decompositions of idempotents of the group algebras of dihedral groups and generalized quaternion groups

1.   Introduction

Given any finite group $G$ and field $F$, denote as $F[G]$ the group ring of $G$ over $F$. When ${\rm char}\, F\nmid|G|$, $F[G]$ is semisimple by Maschke's theorem. Then, by Wedderburn's structure theorem, $F[G]$ is isomorphic to a direct sum of matrix algebras. The Wedderburn decomposition becomes a key tool for studying group algebra problems ^[1,2,3,4,5]. For example, Macedo Ferreira et al. dealt with the Wedderburn $b$-decomposition for alternative baric algebras ^[1]. Jespers et al. reduced the number of generators for a subgroup of finite indexes in a certain kind of unit group $\mathcal{U}(\mathbb{Z}[G])$ by having a closer look at the Wedderburn decomposition of $\mathbb{Q}[G]$ ^[3]. Olivieri et al. studied the automorphism group ${\text{Aut}}(\mathbb{Q}[G])$ of the rational group algebra $\mathbb{Q}[G]$ of a finite metacyclic group $G$ by describing the simple components of the Wedderburn decomposition of $\mathbb{Q}[G]$ ^[5].

As the main objects discussed throughout our paper, dihedral groups $D_{2n}$ describe 2-dimensional objects that have rotational and reflective symmetry, such as regular polygons, and generalized quaternion groups $Q_{4m}$ generalize the quaternion group $Q_8$. In physics, the theory of rigid motion analysis and the practical problem of motion control are all related to quaternions, and many applications in physics use the concept and extension of quaternions.

The Wedderburn decomposition of group algebras of these two families of groups has already attracted much attention. For instance, Giraldo Vergara and Brochero Martínez gave an elementary proof of the Wedderburn decomposition of rational quaternion and dihedral group algebras ^[6]. Giraldo Vergara used the classification of groups of order $\leq32$ and also computed the Wedderburn decomposition of their rational group algebras in order to classify the rational group algebras of dimension $\leq32$ ^[7]. Bakshi et al. calculated a complete set of primitive central idempotents and the Wedderburn decomposition of the rational group algebra of a finite metabelian group ^[8]. Brochero Martínez showed explicitly the primitive central idempotents of $F_{q}[D_{2n}]$ and an isomorphism between the group algebra $F_{q}[D_{2n}]$ and its Wedderburn decomposition when every prime factor of $n$ divides $q-1$ ^[9]. Gao and Yue focused on the algebraic structure of the generalized quaternion group algebras $F_{q}[Q_{4m}]$ over finite field $F_{q}$ ^[10].

Additionally, the study of primitive orthogonal idempotents of group algebras has ignited much interest. For many classes of groups, such as nilpotent, monomial, and supersolvable groups, a complete description of the idempotents of their group algebras was obtained by Berman (see e.g., ^[11]). For example, Berman, in 1995, constructed the minimal central idempotents of the group ring $R(G, F)$ in terms of the central idempotents of $R(H, F)$ when $G$ is an abelian extension of a group $H$. Furthermore, the complete system of minimal idempotents of $R(G, F)$ was given in terms of such a system for $R(H, F)$ when $G/H$ is cyclic ^[12]. After that, he characterized a complete system of primitive orthogonal idempotents of $F[G]$ for any solvable group $G$ of class $M_1$ by calculating linear characters of its subgroups, where $F$ is any field of characteristic prime to $|G|$ containing a primitive root of unity of $|G|$ ^[13].

After nearly 40 years, a method somewhat different but closely related to Berman's in calculating primitive orthogonal idempotents of these group algebras was proposed. In 2004, Olivieri et al. gave a character-free method to describe the primitive central idempotents of $\mathbb{Q}[G]$ when $G$ is a monomial group ^[14]. Later, an explicit and character-free construction of a complete set of primitive orthogonal idempotents of $\mathbb{Q}[G]$ was provided in ^[15] for any finite nilpotent group $G$ (see also ^[16] for the case over finite fields) and in ^[17] for any finite strongly monomial group $G$ such that there exists a complete and non-redundant set of strong Shoda pairs with trivial twistings. See also [18, Chapter 13] for an overall introduction to this topic.

In this paper, after calculating the primitive central idempotents of $\mathbb{C}[D_{2n}]$ and $\mathbb{C}[Q_{4m}]$ via irreducible characters, we further consider their primitive decompositions of idempotents. Note that dihedral groups $D_{2n}$ and generalized quaternion groups $Q_{4m}$ are not only supersolvable groups but also strongly monomial groups. Their primitive decompositions of idempotents can certainly be obtained using Berman's method from ^[13]. Also, a complete set of primitive orthogonal idempotents for any dihedral group can be constructed via strong Shoda pairs, though this is questionable for all generalized quaternion groups [17, § 4]. In contrast, here, the computation of primitive decompositions of idempotents mainly depends on matrix representations of groups and Wedderburn decompositions of group algebras (Lemma 1). Such an approach is theoretically applicable to any semisimple group algebra over an arbitrary field whenever a complete set of its non-equivalent irreducible matrix representations have been obtained. In particular, it is directly available to examples of dihedral groups and generalized quaternion groups.

On the other hand, given two primitive decompositions of idempotents of two isomorphic group algebras, it seems challenging to obtain a specific algebra isomorphism between them that makes the two complete sets of primitive orthogonal idempotents correspond to each other. Here, we solve one small but nontrivial case by establishing an explicit isomorphism between $\mathbb{C}[D_{8}]$ and $\mathbb{C}[Q_{8}]$, which respects the list of primitive orthogonal idempotents we previously found. Indeed, there are plenty of results for the group algebras of $D_{8}$ and $Q_{8}$. For example, Bagiński studied group algebras of 2-groups of maximal class over fields of characteristic $2$, showing that $F_{2}[D_{8}]$ and $F_{2}[Q_{8}]$ are not isomorphic as rings ^[19]. Coleman discussed group rings over the complex and real number fields and over the ring of integers in ^[20], where it was demonstrated that $\mathbb{C}[Q_{8}]\cong \mathbb{C}[D_{8}]$, but $\mathbb{R}[Q_{8}]\ncong \mathbb{R}[D_{8}]$ and $\mathbb{Z}[Q_{8}]\ncong \mathbb{Z}[D_{8}]$. As $\mathbb{R}$ is a field extension of $\mathbb{Q}$, it also implies that $\mathbb{Q}[Q_{8}]\ncong\mathbb{Q}[D_{8}]$. Tambara and Yamagami pointed out that $Q_{8}$ and $D_{8}$ have the same representation ring, but non-isomorphic representation categories as tensor categories ^[21].

Here is the layout of the paper. In Sections 2 and 3, the primitive central idempotents of dihedral groups and generalized quaternion groups are calculated by their irreducible characters. Furthermore, primitive decompositions of idempotents corresponding to their two-dimensional representations are analyzed. In Section 4, two sets of general trigonometric identities reflecting the orthogonality relations of irreducible characters of dihedral groups and generalized quaternion groups are given. In Section 5, a group algebra isomorphism between $\mathbb{C}[Q_{8}]$ and $\mathbb{C}[D_{8}]$ is described, which also provides a correspondence between their primitive orthogonal idempotents previously established.

2.   A primitive decomposition of idempotents of ${{\mathbb C}}[D_{2n}]$

2.1.   Conjugacy classes of $D_{2n}$

Let $D_{2n}$ be the dihedral group of order $2n$, i.e.,

When $n$ is an odd number, namely $n = 2m+1$, $D_{2n}$ has the following conjugacy classes:

When $n$ is an even number, namely $n = 2m$, $D_{2n}$ has the following conjugacy classes:

2.2.   Character table of $D_{2n}$

(i) $n = 2m+1$. We look at the one-dimensional representations first. Note that $D_{2n}/\langle r\rangle \cong \langle s \rangle$, which is abelian, hence the derived subgroup $D'_{2n}\subseteq \langle r\rangle$. Clearly, $s^{-1}r^{-1}sr = r^2\in D'_{2n}$, thus we have $D'_{2n}\supseteq \langle r^2\rangle$. Note that $r^{2m} = r^{-1}\in \langle r^2\rangle$, therefore $\langle r^2\rangle = \langle r\rangle$. Then, $D'_{2n} = \langle r\rangle$. As a result, $D_{2n}$ has two one-dimensional representations and $D_{2n}/\langle r\rangle \cong C_{2}$, where $C_{2}$ is the cyclic group of order 2.

Next, we introduce these two-dimensional irreducible representations of $D_{2n}$ from its natural geometric description [22, Part Ⅰ, 5.3]. We can set up a rectangular coordinate system, where the origin is the center of a regular $n$-sided polygon, and the angular bisector in the first and third quadrants is one of the symmetry axes of the regular $n$-sided polygon. Since $D_{2n}$ is a permutation group of regular $n$-sided polygons, the matrices of $r$, $s$ with respect to the standard basis can be given. Then, we have the following natural representations:

which are $m$ mutually non-equivalent two-dimensional irreducible representations of $D_{2n}$. Thus, when $n$ is an odd number, we set $\theta = \dfrac {2\pi}{n}$, and list the character table of $D_{2n}$ (Table 1):

(ⅱ) $n = 2m$. Similarly, $\langle r^2\rangle$ is a normal subgroup of $D_{2n}$ as $sr^2s^{-1} = r^{-2}\in\langle r^2\rangle$, and $|D_{2n}/\langle r^2\rangle| = 4$, then $D_{2n}/\langle r^2\rangle$ is abelian, and thus $D'_{2n}\subseteq\langle r^2\rangle$. Clearly, $r^2 = s^{-1}r^{-1}sr\in D'_{2n}$, we also have $D'_{2n}\supseteq\langle r^2\rangle$, so $D'_{2n} = \langle r^2\rangle$. As a result, $D_{2n}$ has four one-dimensional representations and $D_{2n}/\langle r^2\rangle\cong C_{2}\times C_{2}$.

If $n$ is an even number, we can also obtain $m-1$ pairwise non-equivalent two-dimensional irreducible representations of $D_{2n}$:

Thus, when $n$ is an even number, we set $\theta = \dfrac {2\pi}{n}$, and list the character table of $D_{2n}$ (Table 2):

2.3.   A primitive decomposition of idempotents

Theorem 2.1. (Wedderburn structure theorem). Let $F$ be any field such that ${\rm char}\, F\nmid|G|$. Then

as algebras, where $D_k$ is a division $F$-algebra, and each matrix algebra $M_{n_{k}}(D_k)$ uniquely determines an irreducible representation $\rho_{k}$ of $G$ up to equivalence, and $n_{k}$ is equal to its dimension over $D_k$ for $k = 1, \dots, s$.

According to Theorem 2.1, we obtain the following useful lemma.

Lemma 2.1. For any semisimple finite group algebra $F[G]$, let $e_{\rho_k}$ be the primitive central idempotent of $F[G]$ corresponding to $\rho_{k}$. The group homomorphism $\rho_k:G\to GL(n_k, D_k)$ can be linearly extended to the following algebra homomorphism

which is an isomorphism when restricted on $F[G]e_{\rho_k}$. In particular, the preimages of the matrix units $E_{11}, \dots, E_{n_k, n_k}$ of $M_{n_{k}}(D_k)$ under this isomorphism provide a primitive decomposition of $e_{\rho_k}$ in $F[G]$. Here, we denote $p_k$ the natural projection.

Also, it is well-known that all primitive central idempotents of the semisimple group algebra $F[G]$ of a finite group $G$ can be obtained by its character table (see e.g., [23, Theorem 3.6.2]), namely

Applying Eq (2.3) to Tables 1 and 2 of dihedral group $D_{2n}$, we immediately have

Proposition 2.1. Let $D_{2n}$ be the dihedral group of order $2n$. The primitive central idempotents corresponding to the one-dimensional irreducible representations of $D_{2n}$ are as follows.

(i) When $n$ is an odd number, namely $n = 2m+1$,

(ii) When $n$ is an even number, namely $n = 2m$,

In order to obtain a primitive decomposition of idempotents of ${{\mathbb C}}[D_{2n}]$, we mainly need to deal with its primitive idempotents corresponding to two-dimensional irreducible representations.

Theorem 2.2. Let $D_{2n}$ be the dihedral group of order $2n$. We have the following primitive decomposition $e_{\rho_{k}} = e'_{\rho_{k}}+e''_{\rho_{k}}$ of the primitive central idempotent $e_{\rho_{k}}$ corresponding to the two-dimensional irreducible representation $({{\mathbb C}}^2, \rho_k)$ of $D_{2n}$ defined in Eqs (2.1) and (2.2) for $k = 1, \dots, \lfloor(n-1)/2\rfloor$.

(i) When $n$ is an odd number, namely $n = 2m+1$,

with $\theta = \dfrac {2\pi}{n}$ and $1\leq k \leq m$;

(ii) When $n$ is an even number, namely $n = 2m$,

with $\theta = \dfrac {2\pi}{n}$ and $1\leq k \leq m-1$.

Proof. Under the group homomorphism $\rho_k:D_{2n}\to GL(2, {{\mathbb C}}), k = 1, \dots, \lfloor(n-1)/2\rfloor$, we have

Therefore,

Thus,

Clearly, $0 < \dfrac{2k\pi}{n} < \pi$, we have

By Lemma 2.1, we know that $F[G]e_{\rho_{k}}\cong M_{n_{k}}(F)$ as algebras, and thus

We can verify that

(ⅰ) If $n = 2m+1$, the primitive central idempotents $e_{\rho_k}$ are given as follows by Eq $\rm (2.3)$ and the character table of $D_{2n}$:

Thus,

Similarly,

(ⅱ) If $n = 2m$, the primitive central idempotents of $D_{2n}$ are given by

Therefore,

Similarly,

□

Example 2.1. Let $D_{8}$ be a dihedral group with order $8$. Then, $m = 2$, $k = 1, n = 4$, there is a primitive decomposition of idempotents as follows.

3.   A primitive decomposition of idempotents of ${{\mathbb C}}[Q_{4m}]$

3.1.   Conjugacy classes of $Q_{4m}$

Let $Q_{4m}$ be the generalized quaternion group of order $4m$, i.e.,

$Q_{4m}$ has the following conjugacy classes:

3.2.   Character table of $Q_{4m}$

The derived subgroup $Q_{4m}^\prime = \langle a^2\rangle$. Indeed, $\langle a^2\rangle$ is a normal subgroup of $Q_{4m}$, and $|Q_{4m}/\langle a^2\rangle| = 4$, hence $Q_{4m}/\langle a^2\rangle$ is abelian and $\langle a^2\rangle\supseteq Q_{4m}^\prime$. Clearly, $b^{-1}a^{-1}b = a$, thus $b^{-1}a^{-1}ba = a^2\in Q_{4m}^\prime$, as $\langle a^2\rangle\subseteq Q_{4m}^\prime$.

As $|Q_{4m}/\langle a^2\rangle| = 4$, $Q_{4m}/\langle a^2\rangle\cong C_{4}$ or $Q_{4m}/\langle a^2\rangle\cong C_{2}\times C_{2}$, and $Q_{4m}$ has four irreducible one-dimensional representations. Also, it has $m-1$ mutually non-equivalent two-dimensional irreducible representations [24, Exs. 17.6, 18.3, 23.5]. We recall these two-dimensional irreducible representations of $Q_{4m}$ as follows.

Let $\varepsilon:= e^{\pi i/m}\in\mathbb{C}$ with $i:= \sqrt{-1}$. For each $k$ with $1\leq k \leq m-1$, denote matrices

which satisfy the following relations:

Hence, it follows that

defined by

is a group homomorphism, and we obtain a representation $({{\mathbb C}}^2, \rho_{k})$ of $Q_{4m}$.

(ⅰ) When $m$ is an odd number, as $2\nmid m$, we know that $b^2 = a^m\not\in Q_{4m}'$, hence the order of $b$ cannot be 2. Then, $b$ is of order 4, so $Q_{4m}/\langle a^2\rangle \cong C_{4}$. We set $\vartheta = \dfrac{\pi}{m}$, and list the character table of $Q_{4m}$ (Table 3):

(ⅱ) When $m$ is an even number, as $2\, |\, m$, we have $b^2 = a^m\in Q_{4m}'$. Therefore, $Q_{4m}/\langle a^2\rangle\cong C_{2}\times C_{2}$. We set $\vartheta = \dfrac{\pi}{m}$, and list the character table of $Q_{4m}$ (Table 4):

3.3.   A primitive decomposition of idempotents

First applying Eq (2.3) to Tables 3 and 4 of generalized quaternion group $Q_{4m}$, we have

Proposition 3.1. Let $Q_{4m}$ be the generalized quaternion group of order $4m$. The primitive central idempotents corresponding to the one-dimensional irreducible representations of $Q_{4m}$ are as follows.

(i) When $m$ is an odd number,

(ii) When $m$ is an even number,

For other primitive idempotents corresponding to two-dimensional irreducible representations of $Q_{4m}$, we have

Theorem 3.1. Let $Q_{4m}$ be the generalized quaternion group of order $4m$. Then, we have the following primitive decomposition $e_{\rho_{k}} = e'_{\rho_{k}}+e''_{\rho_{k}}$ of the primitive central idempotent $e_{\rho_{k}}$ corresponding to the two-dimensional irreducible representation $({{\mathbb C}}^2, \rho_k)$ of $Q_{4m}$ defined in Eq $\rm (3.1)$ for $k = 1, \dots, m-1$.

(i) When $k$ is an odd number,

with $\vartheta = \dfrac {\pi}{m}$ and $1\leq k \leq m-1$;

(ii) When $k$ is an even number,

with $\vartheta = \dfrac {\pi}{m}$ and $1\leq k \leq m-1$.

Proof. (ⅰ) When $k$ is an odd number, under the group homomorphism $\rho_k:Q_{4m}\to GL(2, {{\mathbb C}})$, we have

Then,

Therefore,

As $\varepsilon^{-k}-\varepsilon^{k}\neq0$, it implies that

As a result, we have

Now, one can compute the primitive central idempotents $e_{\rho_{k}}$ via the character table, and then get their desired primitive decompositions by the similar argument as in the proof of Theorem 2.2.

(ⅱ) When $k$ is an even number, by similar arguments as in the case when $k$ is an odd number. □

Example 3.1. Let $Q_{8}$ be a generalized quaternion group of order $8$, then $k = 1$, $m = 2$. Therefore,

4.   Trigonometric identities

Here, we find the following two sets of trigonometric identities covering the orthogonality relations in the character tables of dihedral groups and generalized quaternion groups.

Proposition 4.1. For any $n\geq1$ and $1\leq k \leq n-1$, and any angle $\theta$ that is not an integer multiple of $2\pi$, we have

(i) $\sum\limits_{r = 0}^{n-1} (-1)^r \cos \dfrac{rk\pi}{n} = \begin{cases} 1, & \mathit{{n+k\;odd}}, \\ 0, & \mathit{{n+k\;even}}; \end{cases}$

(ii) $\sum\limits_{r = 1}^{n} \cos r\theta = \dfrac{\sin(\frac{\theta}{2}+n\theta)}{2\sin\frac{\theta}{2}} -\dfrac{1}{2}$.

Proof. (ⅰ) Note that

imply the following product-to-sum identity

As a result, we have

Since $\cos\dfrac{k\pi}{2n}\neq 0$ for any $1\leq k \leq n-1$, we see that

(ⅱ) Similarly by product-to-sum identities, we see that

Since $\theta$ is not an integer multiple of $2\pi$, we obtain that

□

Next, we clarify how these identities are connected to the character tables of dihedral groups and generalized quaternion groups.

Example 4.1. Using the first orthogonality relation in the character tables of $D_{2n}$ in Table 1 when $n = 2m+1$ and $\theta = \dfrac{2\pi}{2m+1}$, we have

The resulting identities

and the identities due to $\langle\chi_{1}, \chi_{\rho_{k}}\rangle = 0$ in Table 2 are all special cases of Prop. 4.1 (ii). Additionally,

That is,

which can also be deduced by Prop. 4.1 (ii).

Example 4.2. Using the first orthogonality relation in the character tables of $Q_{4m}$ in Tables 3 and 4, when $m$ is an odd number,

When $m$ is an even number,

That means

equivalent to Prop 4.1 (i). The identities by $\langle\chi_{3}, \chi_{\rho_{k}}\rangle = 0$ in Table 2 are the same. Also, we have

That is,

which can also be deduced by Prop. 4.1 (ii).

5.   A group algebra isomorphism between $\mathbb{C}[Q_{8}]$ and $\mathbb{C}[D_{8}]$

In this section, we would like to specifically describe a group algebra isomorphism between $\mathbb{C}[Q_{8}]$ and $\mathbb{C}[D_{8}]$, offering a correspondence between two complete sets of their primitive orthogonal idempotents given in Prop. 2.1, Theorem 2.2 and Prop. 3.1, Theorem 3.1.

Theorem 5.1. There is an algebra isomorphism

mapping any $\alpha = x_{0}\cdot{\bf 1}+x_{1}\cdot a^2+x_{2}\cdot a+x_{3}\cdot a^3+x_{4}\cdot b+x_{5}\cdot a^2b+x_{6}\cdot ab+x_{7}\cdot a^3b$ to

with $i:= \sqrt{-1}$ and $x_{i}\in \mathbb{C}$.

Proof. We note that the generalized quaternion group $Q_{4m}$ and the dihedral group $D_{2n}$ have the same character table when $n = 2m$ and $2\, |\, m$. In particular, the smallest case $Q_{8}$ and $D_{8}$ have the same values in the first column, and consequently $\mathbb{C}[Q_{8}]\cong \mathbb{C}[D_{8}]$ as algebras by Lemma 2.1.

The primitive central idempotents corresponding to the two-dimensional irreducible representations of $\mathbb{C}[Q_{8}]$ and $\mathbb{C}[D_{8}]$ are

Under any algebra isomorphism from $\mathbb{C}[Q_{8}]$ to $\mathbb{C}[D_{8}]$, we must have

On the other hand, by Prop. 3.1, all primitive central idempotents corresponding to the four one-dimensional representations of $\mathbb{C}[Q_{8}]$ are as follows:

That is,

By Prop. 2.1, all primitive central idempotents corresponding to the four one-dimensional representations of $\mathbb{C}[D_{8}]$ are as follows:

Namely,

Therefore, we can assume that our desired algebra isomorphism $\psi:\mathbb{C}[Q_{8}]\rightarrow \mathbb{C}[D_{8}]$ satisfies

Furthermore, since $\psi(\xi_{1}\xi_{2}) = \psi(\xi_{1})\psi(\xi_{2})$ for any $\xi_{1}, \xi_{2}\in \mathbb{C}[Q_{8}]$, the map $\psi$ also satisfies:

That is,

as the principal ideal $({\bf 1}-r^2)$ is the annihilator of ${\bf 1}+r^2$. Similarly,

Therefore, we can set

with $k_1, \dots, k_{12}\in \mathbb{C}$, and obtain the following system of equations,

Note that there is more than one solution for this system of equations, and any one of these solutions must also satisfy:

However, these three additional equalities fail to hold simultaneously for any solution in which $k_{1}, k_{5}$, and $k_{9}$ are not all zero. Instead, we find the solution below satisfying all these equations:

That is,

Then

Now, we specifically verify that the stated linear map $\psi:\mathbb{C}[Q_{8}]\rightarrow \mathbb{C}[D_{8}]$ is an algebra isomorphism as desired:

□

According to Prop. 2.1, Theorem 2.2 and Prop. 3.1, Theorem 3.1, we have two complete sets of primitive orthogonal idempotents of ${{\mathbb C}}[Q_{8}]$ and ${{\mathbb C}}[D_{8}]$, respectively. There are primitive idempotents $e_1, \dots, e_4$ corresponding to one-dimensional irreducible representations of $Q_{8}$, and Example 3.1 has calculated the primitive decomposition of idempotents that is given by $\rho_1$ for the unique two-dimensional irreducible representation of $Q_{8}$ up to equivalence.

There are primitive idempotents $\bar{e}_1, \dots, \bar{e}_4$ corresponding to one-dimensional irreducible representations of $D_{8}$, and we see by Example 2.1 that the unique two-dimensional irreducible representation $\rho_1$ of $D_{8}$ up to equivalence provides

Here, we use bar notation to distinguish the complete set of primitive orthogonal idempotents of ${{\mathbb C}}[Q_{8}]$ from that of ${{\mathbb C}}[D_{8}]$.

The proof of Theorem 5.1 has shown that $\psi(e_i) = \bar{e}_i$ for $1\leq i\leq 4$. Now, we further check that

Question. In general, we wonder how to find algebra isomorphisms between $\mathbb{C}[Q_{4m}]$ and $\mathbb{C}[D_{2n}]$ when $n = 2m$ and $2\, |\, m$, making a one-to-one correspondence between the two complete sets of their primitive orthogonal idempotents given in this paper.

6.   Conclusions

Overall, we obtain the formulas for the primitive decompositions of idempotents of the dihedral group algebras $\mathbb{C}[D_{2n}]$ and generalized quaternion group algebras $\mathbb{C}[Q_{4m}]$. Then we present two sets of trigonometric identities by the orthogonality relations of the character tables of these two families of groups. Additionally, we explicitly describe a group algebra isomorphism between $\mathbb{C}[D_{8}]$ and $\mathbb{C}[Q_{8}]$.

Author contributions

Lilan Dai: Writing-original draft and editing, conceptualization, software, methodology; Yunnan Li: Topic selection, writing-review and editing, funding acquisition, methodology, supervision. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

This work is supported by Guangdong Basic and Applied Basic Research Foundation (2022A1515010357).

We would like to thank professor Victor Bovdi for helpful comments.

Conflict of interest

The authors declare that there is no conflict of interest.