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.
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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
xs = x(uH) = vhuH = vhuh-1H = v·huH,
h
-1
where u = huh M. Thus we get an action of G on M that is a combination of
conjugation and multiplication: x = vh acts on u by u →
v·hu .
Lemma 1
The combined action of G on M defined by u →
action of G on S by left multiplication.
v·hu is equivalent with the
Proof
If we define θ:M→S by θ(u) = uH, then as we have already mentioned, θ
becomes a bijection from M to S. Now xθ(u) = x(uH) = vhuH = vhuh-1H = v·huH.
On the other hand θx(u) = θ(vhu ) = θ(v·hu ) = v·hu 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
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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),
we define the monomial of x to be mon(x) = t1α1t2α2 . . . tnαn , 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
ZG,S = |G|-1 Σ mon(g).
g
G
Note that if G has conjugacy classes K1, K2, . . ., Km with gi Ki then
m
ZG,S = |G| Σ | Ki| mon(gi).
-1
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
ZG,S = |G|-1 Σ{mon(uh) | u M, h H}
= |G|-1|H| Σ{mon(u) | u M}
= |M|-1 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}};
mon(1) = t12 , mon(a) = t2 , mon(b) = t12 , and mon(ab) = t2 .
We now have
ZG,S =
(2t12 + 2t2) = (t1 + t2) = ZM,S .
2. Let G = {1, 2, 4, 7, 8, 11, 13, 14}, the group of units mod15.
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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}};
mon(1) = t14 , mon(2) = t4 , mon(4) = t22 , mon(7) = t4, mon(8) = t4,
mon(11) = t14, mon(13) = t4 and mon(14) = t22.
We now have
ZG,S = (2t14 + 2t22 + 4t4) =
(t14 + t22 + 2t4) = 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 EastWest 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