We define a Con-Cos group G to be one having a proper
normal subgroup N whose cosets other than N itself are conjugacy classes. It follows easily that N = G’, the derived group of G. Most of the paper is devoted to trying to classify finite Con-Cos groups satisfying the additional requirement that N has just two conjugacy classes. We show that for such groups the center Z(G) has order at most 2, and if Z(G) = {1}, then G is a Frobenius group of a rather special type.</p

Muktibodh, Arun S.

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Mathematical Journal of OkayamaUniversityVolume 48, Issue 1 2006 Article 8JANUARY 2006Characterization of Frobenius Groups ofSpecial TypeArun S. Muktibodh∗∗Mohota Science CollegeCopyright c©2006 by the authors. Mathematical Journal of Okayama University is produced byThe Berkeley Electronic Press (bepress). http://escholarship.lib.okayama-u.ac.jp/mjouCharacterization of Frobenius Groups ofSpecial TypeArun S. MuktibodhAbstractWe define a Con-Cos group G to be one having a proper normal subgroup N whose cosetsother than N itself are conjugacy classes. It follows easily that N = G’, the derived group of G.Most of the paper is devoted to trying to classify finite Con-Cos groups satisfying the additionalrequirement that N has just two conjugacy classes. We show that for such groups the center Z(G)has order at most 2, and if Z(G) = {1}, then G is a Frobenius group of a rather special type.KEYWORDS: Con-Cos group, 2-Con-Cos groupMath. J. Okayama Univ. 48 (2006), 73{76CHARACTERIZATION OF FROBENIUS GROUPS OFSPECIAL TYPEArun S. MUKTIBODHAbstract. We de¯ne a Con-Cos group G to be one having a propernormal subgroup N whose cosets other than N itself are conjugacyclasses. It follows easily that N = G0, the derived group of G. Most ofthe paper is devoted to trying to classify ¯nite Con-Cos groups satisfy-ing the additional requirement that N has just two conjugacy classes.We show that for such groups the center Z(G) has order at most 2, andif Z(G) = f1g, then G is a Frobenius group of a rather special type.1. IntroductionDe¯nition 1.1. A ¯nite group G is called a Con-Cos group if there existsa proper normal subgroup N in G such that xN = cl(x) for all x 2 G nN ,where cl(x) is the conjugacy class of x.Theorem 1.1. If G is a Con-Cos group and N is the normal subgroup asin the above de¯nition then N = G0, the commutator subgroup of G.Proof. We ¯rst show that G=N is abelian, so that G0 µ N . Let x; y 2 G. Ifx 2 N then xN = N , hence yN commutes with xN . If x is not in N theny¡1xy is not in N . In this case , we have (y¡1N)(xN)(yN) = y¡1xyN =cl(y¡1xy) = cl(x) = xN , so that xN and yN commute with each other.Conversely, let n 2 N and x 2 G nN . Then, xn 2 cl(x), so that there existsy 2 G such that xn = y¡1xy. Hence , n = [x; y], and n 2 G0 ¤Theorem 1.2. Let G be a Con-Cos group. Then G0 = f[x; y] : y 2 Gg, forany x 2 G nG0. In particular, any element of G0 is a commutator.Proof. Each element of G0 is a commutator [x; y] where x is arbitrarily ¯xedin G nG0. ¤Theorem 1.3. If G is a non-abelian Con-Cos group then Z(G) µ G0.Proof. Let a 2 Z(G) and a 6= 1. Then cl(a) = fag. If a is not in G0 thenaG0 = cl(a), so that j G0 j= 1, contrary to the assumption that G is notabelian. ¤Mathematics Subject Classi¯cation. 20Kxx, 20B05.Key words and phrases. Con-Cos group, 2-Con-Cos group.731Muktibodh: Characterization of Frobenius Groups of Special TypeProduced by The Berkeley Electronic Press, 200674 A. S. MUKTIBODH2. 2-Con-Cos groupsDe¯nition 2.1. A ¯nite group G is called a 2-Con-Cos group if the followingconditions are satis¯ed for a proper commutator subgroup G0 of G;i) G0x = cl(x), for all x in G nG0ii) G0 = 1 [ cl(a), for some a in G.Note that, every nonidentity element of G0 is of same order as G0 consistsof only two conjugacy classes. Now, if p is a prime divisor of j G0 j then G0has an element of order p. So, all nonidentity elements of G0 are of orderp. Therefore G0 is a p-group and hence G00 6= G0. Thus G00 is properlycontained in G0. Hence, G00 = 1 as G0 is the minimal normal subgroup of G.Hence G0 is abelian. Thus G0 is elementary abelian.Now, as both G0 and G=G0 are abelian, it immediately follows that G ismetabelian.3. ExamplesSymmetric group S3 of degree 3 is a 2-Con-Cos group with S03 = f1; (123);(123)2g. Alternating group A4 as the set of even permutations on 4 el-ements is a 2-Con-Cos group with A04 = f1; (12)(34); (13)(24); (14)(23)g.Quaternion group Q of order 8 described via generators a; b with relationsa4 = 1; b2 = a2; ba = a¡1b, is a 2-Con-Cos group with Q0 = f1; a2g. Frobe-nius group F42 = Z7 o Z6 with kernel Z7 and complement Z6 ,where Z6acts ¯xed point freely on Z7 n f1g , is a 2-Con-Cos group with G0 = Z7.G = Zp o Aut(Zp) is a 2-Con-Cos group with G0 = Zp. Frobenius groupof order pr(pr ¡ 1) with kernel (Zp)r (elementary abelian) and complementZpr¡1 (cyclic) is a 2-Con-Cos group. However, there do exist Frobeniusgroups of order pr(pr ¡ 1) which are not 2-Con-Cos. e.g. Frobenius groupF702 = Z351 o Z2 with kernel Z351 and complement Z2 where Z2 acts ¯xedpoint freely on Z351nf1g , has order 33(33¡1) . But F702 is not a 2-Con-Cosgroup.4. Some classification theoremsTheorem 4.1. If G is a 2-Con-Cos group such that jG0j 6= 2 then Z(G) = 1.Proof. Let G be a 2-Con-Cos group with jG0j > 2. Then G0 = cl(a) [ f1gwith jcl(a)j > 1. So, if z 2 Z(G) then z = 1 from Z(G) µ G0. ¤Theorem 4.2. If G is a 2-Con-Cos group then jZ(G)j · 2, and if Z(G) =f1g then G is Frobenius with its complement abelian.Proof. jZ(G)j · 2 follows from theorem 4.1. Let G be a 2-Con-Cos groupwith G0 = f1g[ cl(h). Suppose there exists a normal subgroup K in G suchthat 1 6= K . Now, as Z(G) = f1g; 1 6= [G;K] µ G0 \K so jG0 \Kj > 1: As2Mathematical Journal of Okayama University, Vol. 48 [2006], Iss. 1, Art. 8http://escholarship.lib.okayama-u.ac.jp/mjou/vol48/iss1/8CHARACTERIZATION OF FROBENIUS GROUPS OF SPECIAL TYPE 75G0 and K are normal subgroups of G , G0 \K is also a normal subgroup ofG. But G0 is also a minimal normal subgroup of G. Therefore G0 \K = G0.Hence G0 µ K. Thus, G0 is a unique minimal normal subgroup of G. LetjG0j = pm with p a prime and q 6= p be another prime such that q j jGj.Let Q 2 Sylq(G). Then G0Q · G and G0 µ G0Q, so G0Q E G. By Frattiniargument we can write G0NG(Q) = G: Set H = NG(Q), then G0 \H EH:Also G0 \H EG0 as G0 is abelian, and hence G0 \H EG0H = G. By uniqueminimality of G0 we have G0 \ H = f1g or G0 \ H = G0: If G0 \ H = G0then G0 µ H which implies G = G0H = H, hence Q E G. But G0 is theunique minimal normal subgroup of G, so G0 µ Q. Hence p j q, which is acontradiction. Therefore G0 \H = f1g, so H is abelian.Let x 2 H;x 6= 1: Then for any h 2 H;hCG0(x)h¡1 = CG0(x) as H isabelian. But CG0(x)EG0 hence CG0(x)EG0H = G. Therefore CG0(x) = G0or CG0(x) = f1g: If CG0(x) = G0 then x commutes with every element of G0.As H is abelian x 2 Z(G0H) = Z(G) = f1g which is a contradiction. HenceCG0(x) = f1g: This shows that G is a Frobenius group with kernel G0 andcomplement H: ¤Corollary 4.1. If G is a 2-Con-Cos group and G is a Frobenius group withkernel G0 and complement H, then H is cyclic.Proof. Since G is 2-Con-Cos, G0 is elementary abelian. So, H is a group of¯xed point free automorphisms of an abelian group, hence by [5] theoremIX.4.f. H is cyclic. ¤Theorem 4.3. Let G be a 2-Con-Cos group with Z(G) = f1g: Then G is aFrobenius group of the kind (Zp)roZpr¡1 for some prime p and some r ¸ 1.Conversely, Frobenius groups of such kind with kernel (Zp)r and complementZpr¡1 are 2-Con-Cos groups and have trivial center.Proof. G is Frobenius with elementary abelian kernel G0. Complement H iscyclic. HenceG0 = (Zp)r for some prime p andH = Zh for some h ¸ 2. SinceG0 n f1g is a single conjugacy class the action of H on G0 by conjugationhas to be transitive and ¯xed point free, so that jHj = pr ¡ 1. On theother hand, for any Frobenius group G = (Zp)r o Zpr¡1 with kernel (Zp)rand complement Zpr¡1 , G0 is the kernel, Z(G) = f1g and action of thecomplement on (Zp)r n f1g is regular. Hence G is 2-Con-Cos. ¤AcknowledgementUGC has supported this work under the project F. No. 47-111/2002.Thanks to Dr. Leonardo Cangelmi, Italia, Dr. S. G. Dani, Bombay andMr. Vivek Wagh, Nagpur for their valuable comments and suggestions.3Muktibodh: Characterization of Frobenius Groups of Special TypeProduced by The Berkeley Electronic Press, 200676 A. S. MUKTIBODHReferences[1] Mackdonald Ian. D. "Theory of groups" Oxford University Press, 1968.[2] Muktibodh A.S., "Con-Cos groups", Review Bulletin of Calcutta Mathematical So-ciety. Vol.11 , 53{56, 2003.[3] Muktibodh A.S., "Generalized Con-Cos groups", Advances in Algebra,Proceedingsof the ICM Sattelite Conference in Algebra and related topics. 434{441, 2003.[4] Passman D., "Permutation groups", W.A.Benjamin Inc, New York Amsterdam, 1968.[5] Schenkman Eugene, "Group Theory" East West Edition, 1971.Arun S. MuktibodhMohota Science CollegeUmred Rd.Nagpur-440009India.(Received April 25, 2006 )(Revised October 5, 2006 )4Mathematical Journal of Okayama University, Vol. 48 [2006], Iss. 1, Art. 8http://escholarship.lib.okayama-u.ac.jp/mjou/vol48/iss1/8

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