## 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 *r*th root of unity, where \(e=p^fr\) and \(p\not \mid r\). Let

### 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,

### 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

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*(*n*, *k*) 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,

### 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,

### 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

### 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:

### 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

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

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

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

\(\square\)

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:

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

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

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

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

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

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

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.

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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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### Cite this article

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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DOI: https://doi.org/10.1186/s42787-022-00151-0

### Keywords

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

### Mathematics Subject Classification

- Primary 16S34
- Secondary 16U60
- 20C05