Unit groups of finite group algebras of Abelian groups of order 17 to 20

1.   Introduction

Let $FG$ be the group algebra of a finite group $G$ over a finite field $F$ and let $U(FG)$ be the group of units of $FG$. Determining the structure of $U(FG)$ is a classical problem that has already generated considerable interest in the study of group algebra ^[1,10,12,14]. In recent years, units of a group algebra were also used as a tool to tackle many research problems in some other areas including coding theory ^[5,6,7,8] and combinatorial number theory ^[4].

Many researchers have investigated the structure of $U(FG)$ under different conditions. Sandling ^[17] completely determined $U(FG)$ when $G$ is a finite $p$-group and $F$ is a field of characteristic $p$. Creedon ^[2] and Tang et al. ^[19] studied the unit groups of group algebras of some small groups. Tang and Gao ^[18] described the unit group of $FG$ with $|G| = 12$. Maheshwari ^[11] determined the unit group of group algebras $FSL(2, Z_{3})$. Monaghan ^[13] investigated the units of group algebras of non-abelian groups of order $24$ over any finite field of characteristic $3$. Sahai and Ansari ^[15] discussed the unit groups of group algebras of some dihedral groups. In a recent paper, Sahai et al. ^[16] characterized the unit group of $FG$ when $G$ is an abelian group of order at most $16$. In this paper we focus our investigation on the group of units of $FG$ of an abelian group $G$ and determine the structure of $U(FG)$ when $G$ is an abelian group of order between $17$ and $20$.

This paper is organized as follows. In section 2, we provide some preliminary results. Section 3 deals with the unit group of $FG$ when $G$ is a group of prime order (17 or 19). In the last two sections, we determine the structure of $U(FG)$ when $|G| = 18$ and $|G| = 20$, respectively.

2.   Preliminaries

Let $F$ be a finite filed of characteristic $p$ having $q = p^{n}$ elements and $G$ be a finite abelian group. Denote by $C_{n}$ the cyclic group of order $n$ and by $G^{k}$ the direct sum of $k$ copies of an abelian group $G$. Let $F^{n}$ be the direct sum of $n$ copies of $F$ and let $F_{n}$ be the extension of $F$ of degree $n$. Let $V(FG), \; \omega(FG)$, and $J(FG)$ be the group of the normalized unit group, the augmentation ideal and the Jacobson radical of $FG$, respectively. For a subetaoup $H$ of $G$, we denote by $\omega(G, H)$ the left ideal of $FG$ generated by the set $\{h-1 \mid h\in H\}$.

The number of simple components of $FG/J(FG)$ has been given by Ferraz in ^[3]. An element $g\in G$ is called $p$-regular, if $p\nmid o(g)$. In this article we use the same symbols $m$, $\eta$ and $T$ as in ^[3] to represent the least common multiple of the orders of $p$-regular elements of $G$, a primitive $m$th root of unity over the field $F$, and the set

Let $\gamma_{g}$ be the sum of all conjugates of $g\in G$. If $g$ is a $p$-regular element, then the cyclotomic $F$-class of $\gamma_{g}$ is

Lemma 2.1. [3,Proposition 1.2] The number of simple components of $FG/J(FG)$ is equal to the number of cyclotomic $F$-classes in $G$.

Lemma 2.2. [3,Theorem 1.3] Suppose that $Gal(F(\eta)/F)$ is cyclic. Let $t$ be the number of cyclotomic $F$-classes in $G$. If $K_{1}, K_{2}, ..., K_{t}$ are the simple components of $Z(FG/J(FG))$ and $S_{1}, S_{2}, ..., S_{t}$ are the cyclotomic $F$ classes of $G$, then with a suitable re-ordering of indices,

for $i = 1, 2, \ldots, t$.

Remark 2.3. By Lemmas 2.1 and 2.2, we conclude that if $G$ is a finite abelian group and $p\nmid |G|$, then $FG\cong \oplus^t_i K_{i}$, where $K_i$'s are defined in Lemma 2.2.

We also need the following results.

Lemma 2.4. [16,Lemma 4.1] Let $F$ be a finite field of characteristic $p$ with $|F| = q = p^n$ and let $G = C_{pk^{i}}$, where $k, p$ are distinct primes and $i$ is a positive integer. Let $V = 1+J(FG)$. Then

and

Lemma 2.5. [9,Lemma 1.17] Let $G$ be a locally finite $p$-group, and let $F$ be a field of characteristic $p$. Then

Lemma 2.6. [14,Theorem 7.2.7] Let $F$ be a finite field and let $H$ be a normal subetaoup of $G$ with $[G:H] = n < \infty$. Then

If in addition $n\neq0$ in $F$, then

3.   Groups of order 17 and 19

In this section, we describe the structure of $U(FG)$ when the order of the abelian group $G$ is 17 or 19. We need the following two lemmas.

Lemma 3.1. [2,Lemma 4.1] Let $F$ be a finite field of characteristic $p$ with $|F| = q = p^n$, where $p$ is a prime number. Then $U(FC_{p}^{k}) = C_{p}^{np^{k}-n}\times C_{p^{n}-1}$.

Lemma 3.2. [16,Lemma 2.2] Let $F$ be a finite field of characteristic $p$ with $|F| = q = p^n$. If $p\nmid k$, then

Now we can state our first result.

Theorem 3.3. Let $F$ be a finite field of characteristic $p$ with $|F| = q = p^{n}$. Then

Proof. If $p = 17$, applying Lemma 3.1 with $k = 1$, we get

Next we assume that $p \neq 17$. Let $C_{17} = \langle x\rangle.$ Obviously, $m = 17.$

We divide the rest of the proof into several cases according to the value of $q$ module $17$.

Case 1. $q \equiv\pm 1 \mod 17$. By Lemma 3.2, we obtain that

Case 2. $q\equiv\pm2, \pm8\mod17$. It is easy to verify that

By an easy calculation we obtain that

It follows from Remark 2.3 that

So

Case 3. $q\equiv\pm4\mod17$. Then

and thus,

It follows from Remark 2.3 that $FC_{17}\cong F\oplus F_{4}^{4}.$ Therefore,

Case 4. $q\equiv\pm3, \pm5, \pm6, \pm7\mod17$. Then

Thus,

As above, we obtain that $FC_{17}\cong F\oplus F_{16}$, and thus

This completes the proof.

Using a similar method as in the proof of Theorem 3.3, we obtain the following result.

Theorem 3.4. Let $F$ be a finite field of characteristic $p$ with $|F| = q = p^{n}$. Then

4.   Groups of order 18

In this section, we deal with the unit group of $FG$, when $|G| = 18$. Note that if $G$ is an abelian group of 18, then $G\cong C_{18}$ or $G\cong C_{3}\oplus C_{6}$. We need a few lemmas.

Lemma 4.1. ^[2] Let $F$ be a finite field of characteristic $p$ with $|F| = q = p^{n}$. Then

Lemma 4.2. [16,Theorem 3.6] Let $F$ be a finite field of characteristic $p$ with $|F| = q = p^{n}$. Then

Lemma 4.3. [16,Theorem 3.7] Let $F$ be a finite field of characteristic $p$ with $|F| = q = p^{n}$. Then

We now state our result on $U(FC_{18})$.

Theorem 4.4. Let $F$ be a finite field of characteristic $p$ with $|F| = q = p^{n}$. Then

$(1)$ If $p = 2$, then

$(2)$ If $p = 3$, then

$(3)$ If $p\nmid 6$, then

Proof. Let $C_{18} = \langle x\rangle$ and $V = 1+J(FC_{18})$.

(1) If $p = 2$, then applying Lemma 2.4 to $G = C_{18}$, we obtain

and

By Lemma 4.2, we obtain

(2) Suppose $p = 3$. Let $C_{2} = \langle x^9\rangle = \{1, \bar{b}\}$ and $C_{9} = \langle x^2 \rangle = \langle \bar{a}\rangle.$

Note that

By Lemmas 2.5 and 2.6,

and

From the ring epimorphism

we deduce a group epimorphism

and

The ring monomorphism

given by

induces a group monomorphism

And we can verify that $\varphi\sigma = 1_{U(FC_{2})}$. Thus $U(FC_{18})$ is an extension of $U(FC_{2})$ by $V$. So

By Lemma 4.1 we have $U(FC_{2})\cong C_{3^n-1}^{2}.$ We next determine $V$.

Note that

If $\alpha\in J(FC_{18})$, a straight forward computation shows that

and

It follows that $V = 1+J(FC_{18})$ is an abelian $3$-group with exponent dividing $9$. Let

It remains to determine $\ell_1$ and $\ell_2$.

Since $\dim_{F}(V) = \dim_{F}(J(FC_{18})) = \dim_{F}(FC_{18}/FC_{2}) = 16,$ we have $|V| = 3^{16n}$. So $\ell_{1}+2\ell_{2} = 16n$. Let

Then

It follows that $|S| = 3^{4n}$, and thus $\ell_{2} = 4n$. So $\ell_{1} = 8n$ and hence

Therefore,

(3) If $p \nmid 6$, then $m = 18$.

We divide the following proof into several cases according to the value of $q$ module $18$.

Case 3.1. $q\equiv\pm1\mod18$. By Lemma 3.2, we can get

Case 3.2. $q\equiv5, 11\mod18$. Then $T = \{1, 5, 7, 13, 11, 17\}\mod18$. It follows from Remark 2.3 that

Therefore,

So

Case 3.3. $q\equiv7, 13\mod18$. Then $T = \{1, 7, 13\}\mod18$. Thus,

Therefore,

Thus

This completes the proof.

Next we determine the structure of $U(F(C_{3}\times C_{6}))$.

Theorem 4.5. Let $F$ be a finite field of characteristic $p$ with $|F| = q = p^{n}$ and let $G = C_{3}\times C_{6}$.

$(1)$ If $p = 2$, then

$(2)$ If $p = 3$, then

$(3)$ If $p\nmid 6$, then

Proof. Let $G = \langle x, y \mid x^{3} = y^{6} = 1, xy = yx\rangle$ and $V = 1+J(FG)$.

(1) If $p = 2$, then let $H = \langle y^{3}\rangle$. We know that $[G:H] = 9\neq0\in F$. By Lemmas 2.5 and 2.6,

and

From the ring epimorphism

we deduce a group epimorphism

and

The ring monomorphism

induces a group monomorphism

It is not hard to show that $\varphi\sigma = 1_{U(F(C_{3}\times C_{3}))}$. Thus $U(FG)$ is an extension of $U(F(C_{3}\times C_{3}))$ by $V$. So

By Lemma 4.3 we have

We next determine $V$. It is clear that

if and only if

A straight forward calculation gives that $\alpha^{2} = 0$. Thus, it is not hard to show that $\dim_{F}(J(FG)) = 9$, and $V\cong C_{2}^{9n}$. Therefore

(2) If $p = 3$, then let $H = \langle x\rangle\times\langle y^{2}\rangle$. We know $[G:H] = 2\neq0 \in F$. As in the proof of (1) we can show that

By Lemma 4.1 we have $U(FC_{2})\cong C_{3^n-1}^{2}.$ We next determine $V$. It is clear that

It is not hard to show that $\alpha^{3} = 0$. Thus we obtain that $\dim_{F}(J(FG)) = 16$, and $V\cong C_{3}^{16n}$. Therefore,

(3) If $p\nmid 6$, then $m = 6$.

If $q\equiv1\mod6$, then $T = \{1\}\mod6$. Thus,

So

Therefore,

If $q\equiv-1\mod6$, then $T = \{1, 5\}\mod6$. Thus,

Therefore,

Thus

This completes the proof.

5.   Groups of order 20

In this section, we investigate the unit group of $FG$ when $|G| = 20$. Since $G$ is an abelian group of 20, $G\cong C_{20}$ or $G\cong C_{2}\oplus C_{10}$.

Lemma 5.1. [16,Theorem 2.3] Let $F$ be a finite field of characteristic $p$ with $|F| = q = p^{n}$. Then

Lemma 5.2. [16,Theorem 3.1] Let $F$ be a finite field of characteristic $p$ with $|F| = q = p^{n}$. Then

Lemma 5.3. [16,Theorem 3.2] Let $F$ be a finite field of characteristic $p$ with $|F| = q = p^{n}$. Then

The next two theorems provide complete characterizations of the structures of $U(FC_{20})$ and $U(F(C_{2}\oplus C_{10}))$, respectively. As their proofs are very much similar to those of Theorem 4.4 and Theorem 4.5, we omit the detailed computation and state only the results.

Theorem 5.4. Let $F$ be a finite field of characteristic $p$ with $|F| = q = p^{n}$.

$(1)$ If $p = 2$, then

$(2)$ If $p = 5$, then

$(3)$ If $p\neq2$ and $p\neq5$, then

Theorem 5.5. Let $F$ be a finite field of characteristic $p$ with $|F| = q = p^{n}$ and let $G = C_{2}\times C_{10}$.

$(1)$ If $p = 2$, then

$(2)$ If $p = 5$, then

$(3)$ If $p\neq2$ and $p\neq5$, then

Acknowledgments

The authors would like to thank the referees for very useful suggestions and for pointing out the work of R. Sandling to them. This research was supported in part by the Scientific research project of Tianjin Municipal Education Commission (Grant No. 2018KJ252), and was also supported in part by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (Grant No. RGPIN 2017-03093).

Conflict of interest

All authors declare no conflicts of interest in this paper.