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On Unit Group of Finite Semisimple Group Algebras of Non-Metabelian Groups Upto Order 72

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On Unit Group of Finite Semisimple Group Algebras of Non-Metabelian Groups Upto Order 72 ON UNIT GROUP OF FINITE SEMISIMPLE GROUP ALGEBRAS OF NON-METABELIAN GROUPS UPTO ORDER 72 Gaurav Mittal1; R. K. Sharma2 1Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India 2Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, India email: [email protected], [email protected] Abstract. In this paper, we characterize the unit group of semisimple group algebras FqG of k some non-metabelian groups, where Fq is a field with q = p elements for p prime and positive integer k. In particular, we consider all the 6 non-metabelian groups of order 48, only non- metabelian group ((C3 × C3) o C3) o C2 of order 54 and 7 non-metabelian groups of order 72. This completes the study of unit groups of semisimple group algebras for groups upto order 72. Mathematics Subject Classification (2010): 16U60, 20C05 Keywords: Unit group, Finite field, Wedderburn decomposition 1. Introduction k Let Fq denote a finite field with q = p elements for odd prime p, G a finite group and FqG be the group algebra. We refer to [15] for elementary definitions and results related to the group algebras and [2, 17] for the abelian group algebras. One of the most important research problem in the theory of group algebras is determination of their unit groups which are very important from the application point of view, for instance, in the exploration of Lie properties of group algebras, isomorphism problem etc., see [1]. Hurley [7] suggested the construction of convolutional codes from units in group algebra as an important application of units. Talking about some of the existing literature, we refer to [3, 6, 12, 14] for the unit group U(FG) of dihedral groups G and [5, 6, 9, 11, 13, 14, 19, 20, 21] for some non abelian groups other than the dihedral groups. Unit group of finite semisimple group algebras of metabelian groups (groups in which there exists a normal subgroup N of G such that both N and G=N are abelian) has been well studied. From [16], it can be seen that all groups upto order 23 are metabelian. The only non-metabelian groups of order 24 are S4 and SL(2; 3) and their unit group algebras have been discussed in [9, 11]. Further, [16] also implies that there are non-metabelian groups of order 48; 54; 60 and 72. It can be verified that A5 is the only non- metabelian group of order 60 and unit group of its group algebra, i.e. U(FqA5) can be easily deduced from [13] for p ≥ 5. The main motive of this paper is to characterize the unit groups of FqG, where first we consider G to be a non-metabelian group of order 48. There are 6 such groups upto isomorphism. After that we consider the only non-metabelian group of order 54. In the end, we consider all the 1 non-metabelian groups of order 72. In all, we cover the unit groups of 14 semisimple group algebras of non-metabelian groups. Rest of the paper is organized in the following manner: we recall all the basic definitions and results to be used later on in Section 2. Our main results for the characterization of the unit groups are presented in the third, fourth and fifth sections. Some remarks are discussed in the last section. 2. Preliminaries th Let e denote the exponent of G, ζ be a primitive e root of unity and F be an arbitrary finite field. On the lines of [4], we define n IF = fn j ζ 7! ζ is an automorphism of F(ζ) over Fg: Since, the Galois group Gal F(ζ); F is a cyclic group and for any τ 2 Gal F(ζ); F , there exists s a positive integer s which is invertible modulo e such that τ(ζ) = ζ . In other words, IF is ∗ a subgroup of the multiplicative group Ze. For any p-regular element g 2 G, i.e. an element whose order is not divisible by p, let the sum of all conjugates of g be denoted by γg, and the cyclotomic F-class of γg be denoted by S(γg) = fγgn j n 2 IFg: The cardinality of S(γg) and the number of cyclotomic F-classes will be incorporated later on for the characterization of the unit groups. Now, we recall the following two results related to the cyclotomic F-classes. Theorem 2.1. [4] The number of simple components of FG=J(FG) and the number of cyclo- tomic F-classes in G are equal. Theorem 2.2. [4] Let j be the number of cyclotomic F-classes in G. If Ki; 1 ≤ i ≤ j, are the simple components of center of FG=J(FG) and Si; 1 ≤ i ≤ j, are the cyclotomic F-classes in G, then jSij = [Ki : F] for each i after suitable ordering of the indices. For determining the structure of the unit group U(FG), we need Wedderburn decomposition of the group algebra FG. In other words, we want to determine the simple components of FG. Based on the existing literature, we can always claim that F is one of the simple component in the decomposition of FG=J(FG). The simple proof is given here for the completeness. Lemma 2.1. Let A1 and A2 denote the finite dimensional algebras over F. Further, let A2 is ∼ semisimple and g be an onto map between A1 and A2, then we must have A1=J(A1) = A3 + A2, where A3 is some semisimple F-algebra. Proof. From [8], we have J(A1) ⊆ Ker(g). This means there exists F-algebra homomorphism g1 from A1=J(A1) to A2 which is also onto. In other words, we have g1 : A1=J(A1) 7−! A2 defined by g1(a + J(A1)) = g(a); a 2 A1: As A1=J(A1) is semisimple, there exists an ideal ∼ I of A1=J(A1) such that A1=J(A1) = ker(g1) ⊕ I: Our claim is that I = A2. For this to prove, note that any element a 2 A1=J(A1) can be uniquely written as a = a1 + a2, where 2 a1 2 ker(g1); a2 2 I. So, define g2 : A1=J(A1) 7! ker(g1) ⊕ A2 by g2(a) = (a1; g1(a2)): Since, ker(g1) is a semisimple algebra over F, claim and the result holds. Above lemma concludes that F is one of the simple components of FG, provided J(FG) = 0. Now we characterize the set IF defined in the beginning of this section. Theorem 2.3. [10] Let F be a finite field with prime power order q. If e is such that gcd(e; q) = 1, th 2 jq|−1 ζ is primitive e root of unity and jqj is the order of q modulo e, then IF = f1; q; q ; : : : ; q g: Next two results are Propositions 3:6:11 and 3:6:7, respectively, from [15] and are quite useful in our work. Theorem 2.4. If RG is a semisimple group algebra, then RG =∼ RG=G0 ⊕ ∆(G; G0); where G0 is the commutator subgroup of G, RG=G0 is the sum of all commutative simple components of RG, and ∆(G; G0) is the sum of all others. Theorem 2.5. Let RG is a semisimple group algebra and H is a normal subgroup of G. Then RG =∼ RG=H⊕∆(G; H); where ∆(G; H) is left ideal of RG generated by the set fh−1 : h 2 Hg: 3. Unit group of FqG for non-metabelian groups of order 48 The main objective of this section is to characterize the unit groups of FqG, where G is a non- metabelian group of order 48. Upto isomorphism, there are 6 non-metabelian groups of order 48, namely G1 = C2·S4;G2 = GL(2; 3);G3 = A4oC4;G4 = C2×SL(2; 3);G5 = ((C4×C2)oC2)oC3 and G6 = C2 ×S4. Here C2 ·S4 represents the non-split extension of S4 by C2. We consider each of these groups one by one and discuss the unit groups of their respective group algebras along with the WD's in the subsequent subsections (here WD means Wedderburn decomposition and now onwards we use this notation). 3.1. The group G1 = C2 · S4. Group G1 has the following presentation: hx; y; z; w; t j x2t−1; z−1x−1zxt−1w−1z−1; y−1x−1yxy−1; w−1x−1wxt−1w−1z−1; y3; t−1x−1tx; t2; z−1y−1zyw−1z−1; w−1y−1wyt−1z−1; t−1y−1ty; w−1z−1wzt−1; z2t−1; t−1z−1tz; t−1w−1tw; w2t−1i: Also G1 has 8 conjugacy classes as shown in the table below. rep 1 x y z t xz yw xyz order of rep 1 4 3 4 2 8 6 8 where rep means representative of conjugacy class. From above discussion, clearly the exponent 0 ∼ of G1 is 24. Also G1 = SL(2; 3). Next, we give the unit group of FqG1 when p > 3. k Theorem 3.1. The unit group of FqG1, for q = p ; p > 3 where Fq is a finite field having q = pk elements is as follows: 3 (1) for k even or pk 2 f1; 7; 17; 23gmod 24 with k odd ∼ ∗ 2 3 2 U(FqG1) = (Fq) ⊕ GL2(Fq) ⊕ GL3(Fq) ⊕ GL4(Fq): (2) for pk 2 f5; 11; 13; 19gmod 24 with k odd ∼ ∗ 2 2 U(FqG1) = (Fq) ⊕ GL2(Fq) ⊕ GL3(Fq) ⊕ GL4(Fq) ⊕ GL2(Fq2 ): Proof.
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