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Unit group of some finite semisimple group algebras

Abstract

We provide the structure of the unit group of \({\mathbb {F}}_{p^k}S_n\), where \(p>n\) is a prime and \(S_n\) denotes the symmetric group on n letters. We also provide the complete characterization of the unit group of the group algebra \({\mathbb {F}}_{p^k}A_6\) for \(p\ge 7\), where \(A_6\) is the alternating group on 6 letters.

Introduction

Let \(q=p^k\) for some prime p and \(k\in \mathbb {N}\). Let \(\mathbb {F}_q\) denote the finite field of cardinality q. For any group G, let \(\mathbb {F}_qG\) denote the group algebra of G over \(\mathbb {F}_q\). We will follow [14] for basic notations. The group of units of \(\mathbb {F}_qG\) has applications in different areas, including the construction of convolutional codes (see [5,6,7]) and solving problems in combinatorial number theory (see [3]) et cetera. This necessitates finding the explicit structure of the group of units of \(\mathbb {F}_qG\).

In [15], the author has described units of \(\mathbb {F}_qG\), where G is a p-group. The authors of [13] complete the study of unit groups of semisimple group algebras of all groups up to order 120, except that of the symmetric group \(S_5\) and groups of order 96. In Remark 3.7 of this article, we complete the characterization for the group \(S_5\). However, the complexity of the problem increases if the group has a larger size, as that requires solving an equation of type \(\displaystyle \sum \limits _{i=1}^nk_in_i^2=|G|\). The most commonly used tricks include identifying a normal subgroup H of G and considering the algebra \(\mathbb {F}_q(G/H)\) inside \(\mathbb {F}_qG\). This is not possible if the group G does not have any normal subgroup. This is why we will be using representations of the group \(A_6\) to solve the problem in case of the same. See [16, 17] et cetera for more exposition about other groups.

The objective of this article is twofold. We start by investigation of \(\mathbb {F}_qS_n\) where \(p>n\). This is mainly a consequence of the representation theory of \(S_n\) over \({\mathbb {C}}\) and the connection between the Brauer characters of the group when \(p>n\) and the ordinary characters over \({\mathbb {C}}\). We state the characterization in this case in Theorem 3.6. The group of units of the semisimple algebras \(\mathbb {F}_qA_5\) and \(\mathbb {F}_q\text {SL}(3,2)\) have been characterized in [1, 12], respectively. In this article, we look at the next non-Abelian simple group \(A_6\), the alternating group on six letters. We give a complete characterization of \(\mathbb {F}_qA_6\) for the case \(p\ge 7\) in Theorem 4.8.

The rest of the article is organized as follows: in “Preliminaries” section, we give some basic definitions and results. “Units of \(\mathbb {F}_{p^k}S_n\) for \(p \not \mid n\)3” section is about the general description of representations of \(S_n\) over an arbitrary field of characteristic \(p>n\) and deducing the structure of \({\mathcal {U}}(\mathbb {F}_{p^k}S_n)\) for \(p>n\). In section 4, we present the result about \(\mathbb {F}_{p^k}A_6\) where \(p\ge 7\).

Preliminaries

We start by fixing some notations. Already mentioned notations from section 1 are adopted. For a field extension \(E/\mathbb {F}_q\), \(\text {Gal}(E/\mathbb {F}_q)\) will denote the Galois group of the extension. For \(m\in \mathbb {N}\), the notation \(\text {M}(m,R)\) denotes the ring of \(m\times m\) matrices over R and \(\text {GL}(m,R)\) will denote the set of all invertible matrices in \(\text {M}(m,R)\). For a ring R, the set of units of R will be denoted by \(R^\times\). Let Z(R) and J(R) denote the center and the Jacobson radical, respectively. If G is a group and \(x\in G\), then [x] will denote the conjugacy class of x in G. For the group ring \(\mathbb {F}_q G\), the group of units will be denoted as \(\mathcal {U}(\mathbb {F}_q G)\). For the notations on projective spaces, we follow [4].

We say an element g \(\in\) G is a \(p'\)-element if the order of g is not divisible by p. Let e be the exponent of the group G and \(\eta\) be a primitive rth root of unity, where \(e=p^fr\) and \(p\not \mid r\). Let

$$\begin{aligned} I_{\mathbb {F}_q} = \left\{ l~(\text {mod }e):\text { there exists } \sigma \in \text {Gal}(\mathbb {F}_q(\eta )/\mathbb {F}_q)\text { satisfying } \sigma (\eta ) = \eta ^l\right\} . \end{aligned}$$

Definition 2.1

For a \(p'\)-element \(g \in G\), the cyclotomic \(\mathbb {F}_q\)-class of g, denoted by \(S_{\mathbb {F}_q}(\gamma _g)\), is defined as \(\left\{ \gamma _{g^l} : l \in I_{\mathbb {F}_q} \right\}\), where \(\gamma _{g^l}\in \mathbb {F}_q G\) is the sum of all conjugates of \(g^l\) in G.

Then, we have the following results, which are crucial in determining the Artin–Wedderburn decomposition of \(\mathbb {F}_qG\).

Lemma 2.2

[2, Proposition 1.2] The number of simple components of \(\mathbb {F}_qG/J(\mathbb {F}_qG)\) is equal to the number of cyclotomic \(\mathbb {F}_q\)-classes in G.

Definition 2.3

Let \(\pi\) be a representation of a group G over a field F. \(\pi\) is said to be absolutely irreducible if \(\pi ^E\) is irreducible for every field \(F\subseteq E\), where \(\pi ^E\) is the representation \(\pi \otimes E\) over E.

Definition 2.4

A field F is a splitting field for G if every irreducible representation of G over F is absolutely irreducible.

Lemma 2.5

[2, Theorem 1.3] Let n be the number of cyclotomic \(\mathbb {F}_q\)-classes in G. If \(L_1,L_2,\cdots ,L_n\) are the simple components of \(Z(\mathbb {F}_qG/J(\mathbb {F}_qG))\) and \(S_1,S_2,\cdots ,S_n\) are the cyclotomic \(\mathbb {F}_q\)-classes of G, then with a suitable reordering of the indices,

$$\begin{aligned} |S_i| = [L_i:\mathbb {F}_q]. \end{aligned}$$

Lemma 2.6

[11, Lemma 2.5] Let K be a field of characteristic p and let \(A_1\), \(A_2\) be two finite dimensional K-algebras. Assume \(A_1\) to be semisimple. If g : \(A_2\) \(\longrightarrow\) \(A_1\) is a surjective homomorphism of K-algebras, then there exists a semisimple K-algebra l such that \(A_2/J(A_2)\) \(=\) \(l\oplus A_1\).

We need the following lemmas from our previous work to compute some components of the Artin–Wedderburn decomposition of \(\mathbb {F}_qG\), for a finite group G under consideration.

Lemma 2.7

[1, Lemma 3.1] Let G be a group of order n and \(\mathbb {F}\) be a field of characteristic \(p>0\). Let G acts on \(\{1,2,\cdots ,k\}\) doubly transitively. Set \(G_{i}=\{g\in G:g\cdot i=i\}\) and \(G_{i,j}=\{g\in G:g\cdot i=i,g\cdot j=j\}\). Then, the \(\mathbb {F}G\) module

$$\begin{aligned} W=\left\{ x\in \mathbb {F}^k:\displaystyle \sum \limits _{i=1}^kx_i=0\right\} \end{aligned}$$

is an irreducible \(\mathbb {F}G\) module if \(p\not \mid k,p\not \mid |G_{1,2}|\).

Lemma 2.8

[1, Corollary 3.8] Let G be a finite group, K be a finite field of characteristic \(p>0\), \(p\not \mid |G|\). Suppose there exists an n dimensional irreducible representations of G over k. Then, M(nk) appears as one of the components of the Artin-Wedderburn decomposition of the semisimple algebra \(\mathbb {F}_qG\).

Units of \(\mathbb {F}_{p^k}S_n\) for \(p \not \mid n\)

We start the section by talking about representations of \(S_n\) over a finite field. We define the Brauer character and state some important results about representations over an arbitrary field. See [8] for further details.

Let E be a field of characteristic p. We choose a ring of algebraic integers A in \({\mathbb {C}}\) such that \(E = A/M\), where M is a maximal ideal of A containing pA. Take f to be the natural map \(A \longrightarrow E\). Take \(W=\{z\in {\mathbb {C}} | z^m=1\text { for some } m \in {\mathbb {Z}}\text { with } p\not \mid m \}\) (note that \(W \subseteq A\)). Now let \(\pi\) be a representation of a finite group G over E. Let S be the set of \(p'\) elements of G. For \(\alpha \in S\), let \(\epsilon _1,\epsilon _2,\ldots ,\epsilon _l\in E^\times\) be the eigenvalues of \(\pi (\alpha )\) with multiplicities. Then, for every i, there exists a unique \(u_i \in W\) such that \(f(u_i)= \epsilon _i\). Define \(\phi\) : \(S \longrightarrow {\mathbb {C}}\) as \(\phi (\alpha ) = \Sigma u_i\). Then, \(\phi\) is called the Brauer character of G afforded by \(\pi\).

Remark 3.1

The description of Brauer character comes along with a choice of a maximal ideal M of A.

Suppose \(\pi _1,\pi _2,\ldots ,\pi _k\) are all the non-isomorphic irreducible representations of G over E up to isomorphism. Let \(\phi _i\) be the Brauer character afforded by \(\pi _i\). Then, \(\phi _i's\) are called irreducible Brauer characters and we denote by IBr(G) the set {\(\phi _i\)}. We denote by Irr(G) the set of irreducible characters of G over \(\mathbb {C}\). We have the following results.

Lemma 3.2

[8, Theorem 15.13] We have \(IBr(G) = Irr(G)\) whenever \(p\not | |G|\).

For the rest of this section, take \(G= S_n\), the symmetric group on n letters. We say a partition \(\lambda = (\lambda _1,\lambda _2,\cdots ,\lambda _l)\) of n is p-singular if for some j we have \(\lambda _{j+1}=\lambda _{j+2}= \ldots =\lambda _{j+p}\). If a partition is not p-singular, it is called p-regular. Then we have the following.

Lemma 3.3

[9, Theorem 11.5] If F is a field of characteristic p, then as \(\lambda\) varies over the p-regular partitions, \(D^{\lambda }\) varies over the complete set of inequivalent irreducible \(FS_n\)-modules, where \(D^\lambda =\dfrac{S^\lambda }{S^\lambda \cap (S^{\lambda })^{\perp }}\) and \(S^\lambda\) denotes the Specht module corresponding to the partition \(\lambda\). Moreover, every field is a splitting field for \(S_n\).

Proof

The proof follows immediately from the fact that every partition of n is a p-regular partition. \(\square\)

Lemma 3.4

The dimensions of non-isomorphic irreducible representations of \(S_n\) over E coincide with the dimensions of non-isomorphic irreducible representations of \(S_n\) over \(\mathbb {C}\) when characteristic of the field E is greater than n.

Proof

Since the dimension of a representation is as same as the value of the corresponding character \(\chi\) at the identity element of the group, the result follows from Lemma 3.2. \(\square\)

Proposition 3.5

Let \(S_n\) denote the symmetric group on n letters and \(\mathbb {F}_{p^k}\) be a finite field where \(p>n\). Then,

$$\begin{aligned} \mathbb {F}_{p^k}S_n\cong \displaystyle \bigoplus _{\chi \in Irr(G)}\text {M}(\chi (1),\mathbb {F}_{p^k}). \end{aligned}$$

Proof

Since being a semisimple algebra \(\mathbb {C}S_n\cong \displaystyle \bigoplus _{\chi \in Irr(G)}\text {M}(\chi (1),\mathbb {C})\), the result follows from Lemmas 2.8, 3.2 and 3.4. \(\square\)

Theorem 3.6

Let \(S_n\) denote the symmetric group on n letters and \(\mathbb {F}_{p^k}\) be a finite field where \(p>n\). Then,

$$\begin{aligned} \mathcal {U}(\mathbb {F}_{p^k}S_n)\cong \displaystyle \bigoplus _{\chi \in Irr(G)}\text {GL}(\chi (1),\mathbb {F}_{p^k}). \end{aligned}$$

Proof

This follows immediately from Proposition 3.5 and the fact that given two rings \(R_1,R_2\), we have \((R_1\times R_2)^\times =R_1^\times \times R_2^\times\). \(\square\)

Remark 3.7

Theorem 3.6 improves the result of [10] and proves that when \(p>5\), unit group of \(\mathbb {F}_{p^k}S_5\) is \(\mathcal {U}(\mathbb {F}_{p^k}S_5)\) given by

$$\begin{aligned} \mathbb {F}_{p^k}^\times \oplus \mathbb {F}_{p^k}^\times \oplus \text {GL}(4,\mathbb {F}_{p^k})\oplus \text {GL}(4,\mathbb {F}_{p^k})\oplus \text {GL}(5,\mathbb {F}_{p^k}) \oplus \text {GL}(5,\mathbb {F}_{p^k})\oplus \text {GL}(6,\mathbb {F}_{p^k}) \end{aligned}$$

Remark 3.8

For an irreducible representation \(\chi\) of \(S_n\) over a field of characteristic \(p>n\), this is characterized by a partition \(\lambda\) of n. The value \(\chi (1)\) can be calculated as the number of standard Young tableaux of shape \(\lambda\).

Units of \(\mathbb {F}_{p^k}A_6\) for \(p \ge 7\)

We start with the description of the conjugacy classes in \(A_6\). Using [18], the group has 7 conjugacy classes, of which the representatives are given by \((1),a=(1,2)(3,4),b=(1,2,3),c=(1,2,3)(4,5,6),d=(1,2,3,4)(5,6),e=(1,2,3,4,5)\) and \(f=(1,2,3,4,6).\) We have the following relations:

$$\begin{aligned} \text {for all }&g\not \in [e]\cup [f],~[g]=[g^{-1}],\end{aligned}$$
(4.1)
$$\begin{aligned} \text {and }&[e]=[e^4],[e^2]=[e^3]=[f]. \end{aligned}$$
(4.2)

Proposition 4.1

Let \(\mathbb {F}_{q}\) be a field of characteristic \(p\ge 7\) and \(G=A_6\). Then, the Artin–Wedderburn decomposition of \(\mathbb {F}_qG\) is one of the following:

\(\mathbb {F}_q\oplus \bigoplus \limits _{i=1}^6M(n_i,\mathbb {F}_q)\),

\(\mathbb {F}_q\oplus \bigoplus \limits _{i=1}^4M(n_i,\mathbb {F}_q)\oplus M(n_5,\mathbb {F}_{q^2})\)

Proof

Since \(p\ge 7\), we have \(p\not \mid |A_6|\); by Maschke’s theorem we have \(J(\mathbb {F}_qG)=0\) . Hence, Wedderburn decomposition of \(\mathbb {F}_qG\) is isomorphic to \(\bigoplus \limits _{i=1}^nM({n_i},K_i)\), where for all \(1\le i\le n\), we have \(n_i > 0\) and \(K_i\) is a finite extension of \(\mathbb {F}_q\).

Firstly, from Lemma 2.6, we have

$$\begin{aligned} \mathbb {F}_qG \cong \mathbb {F}_q \bigoplus \limits _{i=1}^{n-1}M({n_i},K_i), \end{aligned}$$
(4.3)

taking g to be the map \(g(\sum \limits _{x\in A_6}\alpha _xx)=\sum \limits _{x\in A_6}\alpha _x\). Now to compute these \(n_i\)’s and \(K_i\)’s we calculate the cyclotomic \(\mathbb {F}_q\) classes of G. Note that \(p^k\equiv \pm 1\mod 4,p^k\equiv \pm 1\mod 3\) for any prime p. Hence, \(S_{\mathbb {F}_q}(\gamma _g)=\{\gamma _g\}\) whenever \(g\not \in [e]\cup [f]\) (by Equation 4.1). Hence, we have to consider \(S_{\mathbb {F}_q}(\gamma _g)\) in the other cases.

When \(p\equiv \pm 1\mod 5\), \(S_{\mathbb {F}_q}(\gamma _e)=\{\gamma _e\}\) and \(S_{\mathbb {F}_q}(\gamma _f)=\{\gamma _f\}\), by Eq. 4.2 and the fact that \(p^k\equiv \pm 1\mod 5\). Thus, by Lemmas 2.2 and 2.5, there are seven cyclotomic \(\mathbb {F}_q\)-classes and \([K_i:\mathbb {F}_q]=1\) for all \(1\le i\le 6\). This gives that in this case the Artin–Wedderburn decomposition is

$$\begin{aligned} \mathbb {F}_q\oplus \bigoplus \limits _{i=1}^6M(n_i,\mathbb {F}_q). \end{aligned}$$

When \(p\equiv \pm 2\mod 5\) and k is even, then \(p^k\equiv - 1\mod 5\). Similarly, in this case the Artin–Wedderburn decomposition is

$$\begin{aligned} \mathbb {F}_q\oplus \bigoplus \limits _{i=1}^6M(n_i,\mathbb {F}_q). \end{aligned}$$

Lastly, when \(p\equiv \pm 2\mod 5\) and k is odd, then \(p^k\equiv \pm 2\mod 5\) and \(S_{\mathbb {F}_q}(\gamma _e)=\{\gamma _e,\gamma _f\}\) by Eq. 4.2. Thus, by Lemmas 2.2 and 2.5, there are six cyclotomic \(\mathbb {F}_q\)-classes and \([K_i:\mathbb {F}_q]=1\) for all \(1\le i\le 4\), \([K_5:\mathbb {F}_q]=2\) . In this case, the Artin–Wedderburn decomposition is

$$\begin{aligned} \mathbb {F}_q\oplus \bigoplus \limits _{i=1}^4M(n_i,\mathbb {F}_q)\oplus M(n_5,\mathbb {F}_{q^2}) . \end{aligned}$$

\(\square\)

Since \(\dim \mathbb {F}_{q}A_6=|A_6|=360\), Proposition 4.1 gives that the \(n_i\)’s should satisfy \(n_1^2+n_2^2+n_3^2+n_4^2+n_5^2+n_6^2=359\) or \(n_1^2+n_2^2+n_3^2+n_4^2+2n_5^2=359\). Since these equations do not have a unique solution, we find some of the \(n_i\)’s using representations of \(A_6\) over \(\mathbb {F}_q\) and invoke Lemma 2.7 to reach a unique solution for the mentioned equations. We have the following results.

Lemma 4.2

The group \(S_6\) has four inequivalent irreducible representations of degree 5, which on restriction on \(A_6\) give two inequivalent irreducible representations of \(A_6\) over \(\mathbb {F}_{p^k}\) for \(p\ge 7\). Moreover, these irreducible representations are obtained from two non-isomorphic doubly transitive actions on a set of 6 points.

Proof

Note that \(S_6\) acts on \(T=\{1,2,3,4,5,6\}\) doubly transitively. Hence, by Lemma 2.7, we get an irreducible representation of degree 5. Since tensoring with sign representation gives irreducible representations, we get two inequivalent irreducible representations of degree 5 of \(S_6\), say \(\pi _1\) and \(\pi _2\).

For the other two irreducible representations of dimension 5, we consider the outer automorphism of \(S_6\), say \(\varphi\), given on generators as follows:

$$\begin{aligned} \varphi ((1,2))&=(1,2)(3,4)(5,6)\\ \varphi ((2,3))&=(1,3)(2,5)(4,6)\\ \varphi ((3,4))&=(1,5)(2,6)(3,4)\\ \varphi ((4,5))&=(1,3)(2,4)(5,6)\\ \varphi ((5,6))&=(1,6)(2,5)(3,4). \end{aligned}$$

This gives another doubly transitive action on T, which is not isomorphic to the previous action. Thus, we get another 5 dimensional irreducible representation, say \(\pi _3\). Tensoring \(\pi _3\) with the sign representations, we get \(\pi _4\) which is a 5-dimensional irreducible representation of \(S_6\) different from \(\pi _3\). By considering the characters of the corresponding representations, we see that \(\pi _1,\pi _2,\pi _3\) and \(\pi _4\) are all distinct.

Since \(A_6\) acts doubly transitively on T via the restrictions of these two actions, we obtain two non-isomorphic 5-dimensional irreducible representations of \(A_6\). \(\square\)

Corollary 4.3

The algebra \(\mathbb {F}_qA_6\) has two components to be \(\text {M}(5,\mathbb {F}_q)\) for \(p\ge 7\).

Proof

Immediately follows from Lemmas 4.2 and 2.7. \(\square\)

Corollary 4.4

There does not exist any 4 dimensional irreducible representations of \(A_6\) over \(\mathbb {F}_{p^k}\) for \(p\ge 7\).

Proof

From Lemma 3.3, we know that any field \(\mathbb {F}_{p^k},p\ge 7\) is a splitting field of \(S_6\). Hence, by Proposition 3.5, we have degrees of irreducible representations of \(S_6\) are \(\{1,5,9,10,16\}\).

Recall that by Frobenius reciprocity we have the following bijection

$$\begin{aligned} \text {Hom}_{\mathbb {F}_qS_6}(\text {Ind}V,W)\cong \text {Hom}_{\mathbb {F}_qA_6}(V,\text {Res}W), \end{aligned}$$

where Ind, Res denote the induction functor, restriction functor, respectively. Here V is an irreducible representation of \(A_6\) and W is an irreducible representation of \(S_6\). Suppose \(A_6\) has an irreducible representation V with \(\dim V=4\). Since \([S_6:A_6]=2\), we have that \(\dim \text {Ind}V=8\). Since \(S_6\) does not have any irreducible representation of dimension 8, the induced representation splits. Being \(\dim \text {Ind}V=8\), \(\text {Ind}(V)\) does not have any component of dimensions 9, 10 and 16. Now, let us assume that \(\dim W=5\), then by Lemma 4.2, \(\text {Res}W\) is an irreducible representation. Hence \(\text {Hom}_{\mathbb {F}_qA_6}(V,\text {Res}W)=0\), which implies that \(\text {Ind}V\) does not have any irreducible component of dimension 5. Similarly, \(\text {Ind}V\) does not have any irreducible component of dimension 1. This completes the proof. \(\square\)

Corollary 4.5

The algebra \(\mathbb {F}_qA_6\) has one component to be \(\text {M}(9,\mathbb {F}_q)\) for \(p\ge 7\).

Proof

The group \(A_6\) being isomorphic to \(\text {PSL}(2,\mathbb {F}_9)\) acts doubly transitively on a set with 10 points (see [4]), hence the conclusion. \(\square\)

Corollary 4.6

We have \((n_1,n_2,n_3,n_4,n_5,n_6)=(5,5,9,8,8,10)\) or \((n_1,n_2,n_3,n_4,n_5)=(5,5,9,10,8)\) up to permutation.

Proof

Since \(A_6\) has one 1-dimensional, two 5-dimensional and one 9-dimensional irreducible representations, we can assume that \(n_1=5,n_2=5,n_3=9\). Hence, we are left with the equation

$$\begin{aligned} n_4^2+n_5^2+n_6^2=228\text { or }n_4^2+2n_5^2=228. \end{aligned}$$

Then, \((n_4,n_5,n_6)\in \{(4,4,14),(8,8,10)\},(n_4,n_5)\in \{(14,4),(10,8)\}\). Hence, the result is obvious from Corollary 4.4. \(\square\)

Proposition 4.7

Let \(\mathbb {F}_{p^k}\) be a field of characteristic \(p\ge 7\) and \(A_6\) denote the alternating group on six letters. Then, the Artin–Wedderburn decomposition of \(\mathbb {F}_{p^k}A_6\) is

$$\begin{aligned} \mathbb {F}_q\oplus \text {M}(5,\mathbb {F}_q)\oplus \text {M}(5,\mathbb {F}_q)\oplus \text {M}(9,\mathbb {F}_q)\oplus \text {M}(10,\mathbb {F}_q)\oplus \text {M}(8,\mathbb {F}_{q^2}), \end{aligned}$$

when \(p\equiv \pm 2\mod 5,k\equiv 1\mod 2\) and

$$\begin{aligned} \mathbb {F}_q\oplus \text {M}(5,\mathbb {F}_q)\oplus \text {M}(5,\mathbb {F}_q)\oplus \text {M}(8,\mathbb {F}_q)\oplus \text {M}(8,\mathbb {F}_q)\oplus \text {M}(9,\mathbb {F}_q)\oplus \text {M}(10,\mathbb {F}_q), \end{aligned}$$

otherwise.

Proof

Follows from Proposition 4.1 and Corollary 4.6. \(\square\)

Theorem 4.8

Let \(\mathbb {F}_{p^k}\) be a field of characteristic \(p\ge 7\) and \(A_6\) denote the alternating group on six letters. Then, the unit group of the algebra, \(\mathcal {U}(\mathbb {F}_{p^k}A_6)\) is

$$\begin{aligned} \mathbb {F}_q^\times \oplus \text {GL}(5,\mathbb {F}_q)\oplus \text {GL}(5,\mathbb {F}_q)\oplus \text {GL}(9,\mathbb {F}_q)\oplus \text {GL}(10,\mathbb {F}_q)\oplus \text {GL}(8,\mathbb {F}_{q^2}), \end{aligned}$$
(4.4)

when \(p\equiv \pm 2\mod 5,k\equiv 1\mod 2\) and

$$\begin{aligned} \mathbb {F}_q^\times \oplus \text {GL}(5,\mathbb {F}_q)\oplus \text {GL}(5,\mathbb {F}_q)\oplus \text {GL}(8,\mathbb {F}_q)\oplus \text {GL}(8,\mathbb {F}_q)\oplus \text {GL}(9,\mathbb {F}_q)\oplus \text {GL}(10,\mathbb {F}_q), \end{aligned}$$
(4.5)

otherwise.

Proof

This follows immediately from Proposition 4.7 and the fact that given two rings \(R_1,R_2\), we have \((R_1\times R_2)^\times =R_1^\times \times R_2^\times\). \(\square\)

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Acknowledgements

We thank the referee for a careful reading of the manuscript. This has certainly improved the exposition herein.

Funding

The first named author is partially supported by the IISER Pune research fellowship and the second author has been partially supported by supported by NBHM fellowship

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Correspondence to Saikat Panja.

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Arvind, N., Panja, S. Unit group of some finite semisimple group algebras. J Egypt Math Soc 30, 17 (2022). https://doi.org/10.1186/s42787-022-00151-0

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Keywords

  • Unit group
  • Group algebra
  • Finite field
  • Wedderburn decomposition

Mathematics Subject Classification

  • Primary 16S34
  • Secondary 16U60
  • 20C05