Journal of the Egyptian Mathematical Society (2015) xxx, xxx–xxx
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Units in finite dihedral and quaternion group algebras Neha Makhijani *, R. K. Sharma, J. B. Srivastava Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, India Received 4 March 2014; accepted 2 August 2014
KEYWORDS Group algebra; Unit group; Wedderburn decomposition; Jacobson radical
Abstract Let Fq G be the group algebra of a finite group G over Fq ¼ GFðqÞ. Using the Wedderburn decomposition of F2k D2n =JðF2k D2n Þ, we establish the structure of the unit group of F2k G when G is either D4n , the dihedral group of order 4n or Q4n , the generalized quaternion group of order 4n; n odd. 2010 MATHEMATICS SUBJECT CLASSIFICATION:
Primary: 16U60; Secondary: 16S34; 20C05
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1. Introduction Let FG be the group algebra of a finite group G over a field F and UðFGÞ be its unit group. The study of the group of units is one of the classical topics in group ring theory. Results obtained in this direction are useful for the investigation of Lie properties of group rings, isomorphism problem and other open questions in this area [1]. In [2], Bovdi gave a comprehensive survey of results concerning the group of units of a modular group algebra of characteristic p. There is a long tradition on the study of the unit group of finite group algebras [3–12]. In general, the structure of UðFGÞ is elusive if jGj ¼ 0 in F. * Corresponding author. E-mail addresses:
[email protected] (N. Makhijani),
[email protected] (R.K. Sharma),
[email protected] (J.B. Srivastava). Peer review under responsibility of Egyptian Mathematical Society.
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Let us introduce the background of our investigation. The structure of UðF2 D2p Þ was determined by Kaur and Khan in [13] for an odd prime p. Recently, the authors generalized this result and computed the structure of the unit group of F2k D2n when n is odd. In this note, we use the Wedderburn decomposition of F2k D2n =JðF2k D2n Þ obtained in [14] to study the unit group of F2k D4n and F2k Q4n when n is odd. In what follows, q ¼ 2k ; ordl ðmÞ denotes the multiplicative order of m modulo l when ðl; mÞ ¼ 1 and uðnÞ denotes the Euler’s phi function on a positive integer n. 2. Main Results In this section, we begin by considering the lemmas that are essential for developing the proof of main results. Lemma 2.1. Let F be a perfect field, G be a finite group and JðFGÞ be the Jacobson radical of FG. Then UðFGÞ ffi ð1 þ JðFGÞÞ U ðFG=JðFGÞÞ
http://dx.doi.org/10.1016/j.joems.2014.08.001 1110-256X ª 2015 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. Please cite this article in press as: N. Makhijani et al., Units in finite dihedral and quaternion group algebras, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.08.001
2
N. Makhijani et al.
Proof. Observe that
Fq D2n =JðFq D2n Þ ffi Fq w
inc
uðmÞ
Mð2; Fqem Þ 2em
mjn; m>1
1!1 þ JðFGÞ ! UðFGÞ ! U ðFG=JðFGÞÞ!1
Now
is a short exact sequence of groups, where wðxÞ ¼ xþ JðFGÞ 8 x 2 UðFGÞ. By Wedderburn–Malcev theorem [15, Thm. 6.2.1], it follows that there exists a semisimple subalgebra B of FG such that
DðD2N ; hAn iÞ ¼ DðhAn iÞFq D2N ¼ JðFq hAn iÞFq D2N # JðFq D2N Þ showing that dimFq J Fq D2N P 2N 2n. Since D2N =hAn i ffi D2n , there exists an onto Fq -algebra homomorphism
FG ¼ B JðFGÞ
/ : Fq D2N ! Fq D2n =J Fq D2n given by the assignment A #a þ J Fq D2N ; B #b þ J Fq D2N whence J Fq D2N # ker / and dimFq J Fq D2N 6 2N ð2n 1Þ ¼ 2N 2n þ 1
and thus for each x þ JðFGÞ 2 FG=JðFGÞ, there exists a unique xB 2 B such that x þ JðFGÞ ¼ xB þ JðFGÞ Define h : U ðFG=JðFGÞÞ ! UðFGÞ as hðx þ JðFGÞÞ ¼ xB
8 x þ JðFGÞ 2 UðFG=JðFGÞÞ
Then h is a group homomorphism such that woh ¼ id jU ðFG=JðFGÞÞ and hence UðFGÞ ffi ð1 þ JðFGÞÞ U ðFG=JðFGÞÞ For a normal subgroup H of G, the natural homomorphism eH : G ! G=H can be extended to an F-algebra epimorphism eH : FG ! FðG=HÞ. The kernel of eH is denoted by DðG; HÞ and DðGÞ ¼ DðG; GÞ. Lemma 2.2 [16, Lemma 1.17]. Let G be a locally finite p-group and F be a field of characteristic p. Then JðFGÞ ¼ DðGÞ. Lemma 2.3 [17, Ch. 1, Prop. 6.16]. Let f : R1 ! R2 be a surjective homomorphism of rings. Then
But there is only one 1-dimensional representation of D2N over Fq . This proves that dimFq J Fq D2N ¼ 2N 2n þ 1 and J Fq D2N ¼ ker / giving Fq D2N =J Fq D2N ffi Fq D2n =J Fq D2n The decomposition of Fq Q4N =J Fq Q4N working on parallel lines. h
UðFq D4n Þ ffi
ð2nþ1Þk C2
where (
Lemma 2.4 [18, Theorem 7.2.7]. Let H be a normal subgroup of G with ½G : H ¼ n < 1. Then JðFGÞn # JðFHÞFG # JðFGÞ. If in addition n – 0 in F, then JðFGÞ ¼ JðFHÞFG.
Proof. Let
ffi Fq where ( em ¼
uðmÞ
Mð2; Fqem Þ 2em
mjn; m>1
Cq1
GLð2; Fqem Þ
uðmÞ 2em
!
dm =2 if dm is even and qdm =2 1mod m dm otherwise
and dm ¼ ordm ðqÞ.
Fq Q4N =JðFq Q4N Þ ffi Fq D2N =JðFq D2N Þ
Y
mjn; m>1
with equality if ker f # JðR1 Þ.
Lemma 2.5. Let N ¼ 2t n such that 2-n. Then
can be obtained by
Theorem 2.6. If n is odd, then
em ¼
fðJðR1 ÞÞ # JðR2 Þ
D4n ¼ h a; b j a2n ; b2 ; b1 ab ¼ a1 i and X ¼ 1 þ an and then fX;aX; ;an1 X;bX; baX; ;ban1 Xg is a basis of DðD4n ; han iÞ. Observe that any W 2 DðD4n ; han iÞ is expressible as W ¼ A1 þ A2 a þ þ An an1 þ Anþ1 b þ Anþ2 ba þ þ A2n ban1 X
for some Ai 2 Fq so that
dm =2 dm
if dm is even and qdm =2 1 mod m otherwise
W2 ¼ A1 þ A2 a þ þ An an1 þ Anþ1 b 2 þ Anþ2 ba þ þ A2n ban1 ð1 þ an Þ2 ¼ 0 That is, 1 þ DðD4n ; han iÞ ffi C2nk 2 . The Fq algebra homomorphism
and dm ¼ ordm ðqÞ. Proof. To distinguish the elements of D2N from those of D2n , let D2N be presented by hA; B j AN ; B2 ; B1 AB ¼ A1 i and D2n by ha; b j an ; b2 ; b1 ab ¼ a1 i From [14], it is known that
h : Fq D4n ! Fq D2n =JðFq D2n Þ given by the assignment a #a þ JðFq D2n Þ; b #b þ JðFq D2n Þ is onto. d Thus if B ¼ It is known that D 2n 2 JðFq D2n Þ. ð1 þ a þ þ an1 Þð1 þ bÞ, then hðBÞ ¼ 0 þ JðFq D2n Þ showing
Please cite this article in press as: N. Makhijani et al., Units in finite dihedral and quaternion group algebras, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.08.001
Units in finite dihedral and quaternion group algebras
3
that B 2 kerh ¼ JðFq D4n Þ. In fact, JðFq D4n Þ ¼ DðD4n ; han iÞ Fq B as a vector space over Fq . Since B2 ¼ ðð1 þ bÞð1 þ a þ þ an1 ÞÞ nþ1
¼ ð1 þ bÞð1 þ ba nþ1
¼ ð1 þ b þ a
2
n1 2
H¼
Þ 1 þ a þ þ a nþ1
þ ba
Notice that if Y ¼ 1 þ Cn , then fY; CY; ; Cn2 Y; DY; b ð1 þ DÞ C; b Ug is a basis of JðFq Q4n Þ over DCY; ; DCn2 Y; C; Fq . Since Y 2 ZðFq Q4n Þ and Y2 ¼ 0, it is therefore evident that
2
2n2
Þ 1 þ a þ þ a
(
) ! n2 n2 X X i i Bi DC Y Ai ; Bi 2 Fq Ai C þ
1þ
i¼0
i¼0
6 1 þ JðFq Q4n Þ
¼ 1 þ a2 þ þ a2n2 þ b þ ba2 þ þ ba2n2 þ anþ1
ð2n2Þk
and H ffi C2
. n o b þ A3 ð1 þ DÞ C b j Ai 2 Fq 6 1þ Also K ¼ 1 þ A1 C þ A2 C
þ anþ3 þ þ a3n1 þ banþ1 þ banþ3 þ þ ba3n1 ¼ 0 because n is odd and a2n ¼ 1:
JðFq Q4n Þ and by Lemma 2.7, we find K ffi Ck2 Ck4 .
and
Since H \ K ¼ f1g and j1 þ JðFq Q4n Þj ¼ jHKj, it follows that 1 þ JðFq Q4n Þ ¼ HoK. This completes the proof. h
a ð1 þ bÞ; XB ¼ ð1 þ an Þð1 þ a þ þ an1 Þð1 þ bÞ ¼ b we find that WB ¼ BW so that
V ¼ 1 þ JðFq D4n Þ ¼ 1 þ DðD4n ; han iÞ f1 þ gB j g 2 Fq g
References
ð2nþ1Þk
ffi C2
This completes the proof.
h
A group G is said to be a general product of its subgroups L and M if G ¼ LM; L \ M ¼ f1g In this case, we write G ¼ LoM. In the subsequent theorem, it is established that 1 þ JðFq Q4n Þ is a general product of two of its proper subgroups. As a consequence, the structure of UðFq Q4n Þ is obtained. Lemma 2.7 [3, Lemma 1.1]. Let G be a finite abelian p-group, k Q i i i Cnpii , then Gp ¼ fxp j x 2 Gg and pmi ¼ jGp j. If G ffi i¼1
ni ¼ mi1 2mi þ miþ1
8 i; 1 i k
Theorem 2.8. If n is odd, then UðFq Q4n Þ ffi
ð2n2Þk C2 o
Ck2 Ck4
o Cq1
Y
uðmÞ 2em
GLð2; Fqem Þ
mjn; m>1
!
where em as in Theorem 2.6. Proof. Let Q4n be presented by hC; D j C2n ; D2 ¼ Cn ; D1 CD ¼ C1 i and U ¼ ð1 þ DÞð1 þ C þ . . . þ Cn1 Þ. Then via similar arguments as in the previous theorem, U 2 JðFq Q4n Þ and b 1 þ C þ . . . þ Cn1 U 2 ¼ ð1 þ D Þð1 þ C n Þ þ D C
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b þ DC b¼C b ¼ ð1 þ D Þ C
Please cite this article in press as: N. Makhijani et al., Units in finite dihedral and quaternion group algebras, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.08.001
