International Mathematical Forum, Vol. 7, 2012, no. 30, 1491 - 1494

Cycle Index of Internal Direct Product Groups

I. N. Kamuti

Mathematics Department, Kenyatta University P. O. Box 43844 –00100, Nairobi, Kenya [email protected]

Abstract

If M and H are permutation groups with cycle indices ZM and ZH respectively, and if  is some binary operation on permutation groups, then a fundamental problem in enumerative combinatorics is the determination of a formula for ZM  H in terms of ZM and ZH. To this end, a number of results have already been obtained (cf. Harary [1], [2], [3]; Harary and Palmer [6]; Harrison and High [7]; Pόlya [10]). This paper may be viewed as a continuation of a previous paper (Kamuti [8]) in which I have shown how the cycle index of a semidirect product group G= M H can be expressed in terms of the cycle indices of M and H by considering semidirect products called Frobenius groups. Thus if G=M H (internal direct product), the aim of this paper is to express the cycle index of G in terms of the cycle indices of M and H when G acts on the cosets of H in G.

Mathematics Subject Classification: 05A19

Keywords: Internal direct product, Equivalent actions, Cycle indices.

INTRODUCTION

The cycle index of external direct product of two groups may be found in a number of books and articles (Harary [5], [4] p. 184; Krishnamurthy [9] p. 146). The cycle index of semidirect products is given by Kamuti [8] by considering particularly useful semidirect product groups; namely Frobenius groups. This paper is devoted to a very special case of semidirect products called internal direct product.

1492 I. N. Kamuti

ISOMORPHISM OF PERMUTATION GROUPS

Let (G1, S1) and (G2, S2) be permutations groups. To say that (G1,S1) (G2,S2) (permutation isomorphism) means there exists a group isomorphism Φ : G1 → G2 and a bijection θ : S1 → S2 so that θ (xs) = Φx (θs) for all x G1, s S1, or θ x = Φx θ for all x G1. An important special case is when G1 = G2, and Φ is the identity map. Then the condition is θ x = x θ for all x G, and the definition determines the notion of equivalent actions of G on two sets S1 and S2.

EQUIVALENT ACTIONS OF INTERNAL DIRECT PRODUCTS

Suppose G = M H (internal direct product), M G, H G, MH = G and M H = 1. Then G acts on S = G/H, set of left cosets of H in G by left multiplication, that is if x G, yH S, then x(yH) = xyH S. Note that |S| =|G|/|H| = |M|, and that G also acts on M by conjugation since M G. Furthermore there is a natural bijection between M and S, given by u → uH for each u M. However, that does not determine equivalent actions of G on S and M. We may work out a more complicated action of G on M, which is equivalent to its action on S. Each x G can be written (uniquely) as x = vh, with v M, h H. Again each s S can be written (also uniquely) as s = uH, with u M. We have -1 h xs = x(uH) = vhuH = vhuh H = v· uH, h -1 where u = huh M. Thus we get an action of G on M that is a combination of h conjugation and multiplication: x = vh acts on u by u → v· u .

Lemma 1 h The combined action of G on M defined by u → v· u is equivalent with the action of G on S by left multiplication.

Proof If we define θ:M→S by θ(u) = uH, then as we have already mentioned, θ -1 h becomes a bijection from M to S. Now xθ(u) = x(uH) = vhuH = vhuh H = v· uH. h h On the other hand θx(u) = θ(vhu ) = θ(v· u ) = v· u H. Thus x θ = θ x.

DERIVATION OF THE CYCLE INDEX OF INTERNAL DIRECT PRODUCTS

We now compute the cycle index of internal direct products, beginning with a general discussion of cycle indices.

Definition 1 If a finite group G acts on a set S with n elements, each x G corresponds to a permutation σ of S, which can be written uniquely as a product of disjoint cycles. Cycle index of internal direct product groups 1493

If σ has α1 cycles of length 1, α2 cycles of length 2, . . . , αn cycles of length n, we say that σ and hence x has cycle type (α1, α2, . . . , αn).

Definition 2 If a finite group G acts on a set S, |S| = n, and x G has cycle type (α1, α2, . . . , αn), α1 α2 αn we define the monomial of x to be mon(x) = t1 t2 . . . tn , where t1, t2, . . . , tn are distinct commuting indeterminates.

Definition 3 The cycle index of the action of G on S is the polynomial (say over the rational field Q) in t1, t2, . . . , tn given by

-1 ZG,S = |G| Σ mon(g). g G

Note that if G has conjugacy classes K1, K2, . . ., Km with gi Ki then m -1 ZG,S = |G| Σ | Ki| mon(gi). i=1

Next we look at the cycle index of G = M  H, internal direct product. Let x G, then from the previous section x can be written (uniquely) as x = uh, with u M, h H. Also each s S = G/H can be written uniquely as s = uH with u M. Again from the previous section the action of x = uh on v M becomes: v → u.hv = uv (since elements of M and H commute).Thus mon(uh) = mon(u) for all u and h. So

-1 ZG,S = |G| Σ{mon(uh) | u M, h H} -1 = |G| |H| Σ{mon(u) | u M} -1 = |M| Σ{mon(u) | u M}

= ZM,S .

Examples 1. Let G = {1, a, b, ab}, the Klein 4-group. Take M = {1, a} and H = (1, b), then G = M H. Now G/H = S = {{1, b}, {a, ab}}; 2 2 mon(1) = t1 , mon(a) = t2 , mon(b) = t1 , and mon(ab) = t2 .

We now have

= 2t 2 + 2t = t + t = ZG,S ( 1 2) ( 1 2) ZM,S .

2. Let G = {1, 2, 4, 7, 8, 11, 13, 14}, the group of units mod15. 1494 I. N. Kamuti

Take M = <2> = {1, 2, 4, 8} and H = <11> = {1, 11}, then G = M H. Now G/H = S = {{1, 11}, {2, 7},{4, 14}, {8, 13}}; 4 2 mon(1) = t1 , mon(2) = t4 , mon(4) = t2 , mon(7) = t4, mon(8) = t4, 4 2 mon(11) = t1 , mon(13) = t4 and mon(14) = t2 . We now have

= 2t 4 + 2t 2 + 4t = t 4 + t 2 + 2t = ZG,S ( 1 2 4) ( 1 2 4) ZM,S .

REFERENCES

[1] F. Harary, On the number of bi-coloured graphs, Pacific Journal of Mathematics, 8 (1958), 743-755.

[2] F. Harary, Exponentiation of permutation groups, The American Mathematical Monthly, 66 (1959), 572-575.

[3] F. Harary, Applications of Pόlya’s theorem to permutation groups, A seminar on graph theory, Chapter 5, Holt, Rinehart and Winston, New York, 1967.

[4] F. Harary, Graph theory, Addison-Wesley Publishing Company, New York, 1969.

[5] F. Harary, Enumeration under group action: Unsolved graphical enumeration problems, IV, Journal of Combinatorial Theory, 8 (1970), 1 - 11.

[6] F. Harary, and E. Palmer, The power group enumeration theorem, Journal of Combinatorial Theory, 1 (1966), 157-173.

[7] M. A. Harrison, and R. G. High, On the cycle index of a product of permutation groups, Journal of Combinatorial Theory, 4 (1968), 277-299.

[8] I. N. Kamuti, On the cycle index of Frobenius groups, East African Journal of Physical Sciences, 5(2) (2004), 81-84.

[9] V. Krishnamurthy, Combinatorics, theory and applications, Affiliated East- West Press Private Limited, New-Delhi, 1985.

[10] G. Pόlya, Kombinatorische anzahlbestimmungen für gruppen, grappen, und chemische verbindugen, Acta Mathematica, 68 (1937), 145-253.

Received: December, 2011