Derivation of the Cycle Index Formula of the Affine Square(Q)

International Journal of Algebra, Vol. 13, 2019, no. 7, 307 - 314 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2019.9725

Derivation of the Cycle Index Formula of the Aﬃne Square(q) Group Acting on GF (q)

Geoﬀrey Ngovi Muthoka

Pure and Applied Sciences Department Kirinyaga University, P. O. Box 143-10300, Kerugoya, Kenya

Ireri Kamuti

Department of Mathematics and Actuarial Science Kenyatta University, P. O. Box 43844-00100, Nairobi, Kenya

Mutie Kavila

Department of Mathematics and Actuarial Science Kenyatta University, P. O. Box 43844-00100, Nairobi, Kenya

This article is distributed under the Creative Commons by-nc-nd Attribution License. Copyright c 2019 Hikari Ltd.

Abstract

The concept of the cycle index of a permutation group was discovered by [7] and he gave it its present name. Since then cycle index formulas of several groups have been studied by different scholars. In this study the cycle index formula of an affine square(q) group acting on the elements of GF (q) where q is a power of a prime p has been derived using a method devised by [4]. We further express its cycle index formula in terms of the cycle index formulas of the elementary abelian group Pq and the cyclic group C q−1 since the affine square(q) group is a semidirect 2 product group of the two groups.

Keywords: Cycle index, Aﬃne square(q) group, Semidirect product group 308 Geoﬀrey Ngovi Muthoka, Ireri Kamuti and Mutie Kavila

1 Introduction

[1] The set Pq = {x + b, where b ∈ GF (q)} forms a normal subgroup of the affine(q) group and the set C q−1 = {ax, where a is a non zero square in GF (q)} 2 forms a cyclic subgroup of the affine(q) group under multiplication. The semidirect product Pq C q−1 form a group known as the affine square(q) o 2 group denoted by Aff(q). The elements of Aff(q) are of the form {ax + b such that b ∈ GF (q) and a is a non zero square in GF (q)}. In this study we derive the cycle index formula of the affine square(p) group acting on GF (q) and express it in terms of the cycle index formulas of the elementary abelian group Pq and the cyclic group C q−1 . 2

2 Preliminary Notes

Deﬁnition 2.1. [2] A group G is said to be a semidirect product group of N by H if; N C G and H < G, N ∩ H = {e} and NH = G. We symbolically express this as G = N o H.

Deﬁnition 2.2. [4] The M¨obiusfunction of any n ∈ N is given by,  −1 if n is a square free with an odd number of prime factors  µ(n) = 0 if n has a squared prime factor 1 if n is a square free with an even number of prime factors.

Deﬁnition 2.3. [3] The cycle index of the action of G on X is the polynomial (say over the rational ﬁeld Q) in t1, t2, . . ., tn given by;

m 1 X Z(G) = Z (t , t , ··· , t ) = {mon(g)} G,x 1 2 n |G| g∈G

Note that if G has conjugacy classes K1,K2, . . ., Km with gi ∈ Ki then

m 1 X Z(G) = |K | mon(g ). |G| i i i=1

Theorem 2.4. [6] Let x be a permutation with cycle type (α1, α2, α3, . . ., αn) then;

l l P i. the number π(x ) of cycles of length one in x is i|l iαi.

i i P l ii. αl = l i|l π x µ(i) where µ is the M¨obiusfunction. The cycle index of the aﬃne square(q) group 309

3 Main Results

3.1 The cycle index of the aﬃne square(q) group

Lemma 3.1. Let g ∈ Aff(q) be such that g ﬁxes only one element in GF (q), then CG(g) = C q−1 . 2

The elements of Aff(q) which ﬁx 0 ∈ GF (q) are of the form αx + 0 where α 0 α ∈ GF (q). This can be written as; M = . 0 1 a b A general element of Aff (q) is of the form such that a is an non 0 1 a b −1 1 − b zero square element of GF (q) and = a a . 0 1 0 1 a b α 0 1 − b α −αb + b Now a a = . 0 1 0 1 0 1 0 1 a b α 0 For the element to centralize then, −αb + b = 0, implying 0 1 0 1 b(1 − α) = 0 ⇒ b = 0 or α = 1. α 0 If α = 1, then is the identity which is centralized by every element 0 1 α 0 a 0 of the group and so the centralizers of are of the form 0 1 0 1 where a is a non zero square element in GF (q) thus the centralizer has order (q−1) (q−1) 2 which is a cyclic multiplicative group C (q−1) of order 2 . 2

Lemma 3.2. Let g ∈ Aff(q) be such that g does not ﬁx any element in GF (q), then CG(g) = Pq.

Proof. Since the Affine square(q) group acts transitively on GF (q) [5], then the stabilizers of each of the elements in GF (q) are conjugate and only intersect at the identity so it is enough to find the centralizer of any one element. The elements of Aff(q) which do not fix any element of GF (q) are of the form 1 α x + α where α ∈ GF (q). This can be written as; M = . 0 1 a b A general element of Aff (q) is of the form such that a is a non 0 1 a b −1 1 − b zero square element in GF (q) and = a a . 0 1 0 1 310 Geoffrey Ngovi Muthoka, Ireri Kamuti and Mutie Kavila

a b 1 α 1 − b 1 αa Now a a = . 0 1 0 1 0 1 0 1 a b 1 α For the element to centralize then, αa = α, implying 0 1 0 1 α(a − 1) = 0 ⇒ α = 0 or a = 1. 1 α If α = 0, then is the identity which is centralized by every element of 0 1 1 α 1 b the group and so the centralizers of are of the form where 0 1 0 1 b ∈ GF (q) thus the centralizer has order q which is the elementary abelian group Pq. Theorem 3.3. Let p > 2 be a prime and q = pr, the cycle index formula of the affine square(q) group G acting on the q elements of GF (q) is given by;   1 q X q−1 Z = tq + (q − 1) t p + q ∅(d)t t d , (G,X) |G|  1 p 1 d  q−1 16=d| 2 1 where |G| = 2 q(q − 1), ∅(d) is the Euler’s phi function and X the q elements of the GF (q) . Proof. The elements of the Aff (q) group are partitioned into I, τ (the set of 1 elements that fix one element on the field GF (q)) and τ0 (the set of elements that do not fix any element of GF (q)). To derive the cycle index formula we need to find the number of τ0 and τ1 elements and the respective cycle types. Let g ∈ τ1. Then from Lemma 3.1 CG(g) = C q−1 . 2 1 g 2 q(q − 1) ⇒ |C | = 1 = q (1) 2 (q − 1) where Cg is the conjugacy class in G containing g. Let g ∈ τ0, then from Lemma 3.2 CG(g) = Pq

1 q(q − 1) q − 1 ⇒ |Cg| = 2 = (2) q 2 1 2 q(q−1) The subgroup NG C q−1 = C q−1 , implying there are 1 = q conjugate 2 2 2 (q−1) cyclic groups C q−1 in G. 2 These cyclic groups intersect only at the identity. Thus;

q − 1 |τ | = − 1 q (3) 1 2 The cycle index of the aﬃne square(q) group 311

We ﬁnd the number of τ0 elements by subtracting the number of τ1 elements and the identity from the order of G. It follows that;

q − 1 q − 1 |τ | = q − − 1 q − 1 = q − 1 (4) 0 2 2

q−1 Since the length of a conjugacy class of g ∈ τ0 is 2 from Equation 2 and the number of τ0 elements are q − 1 from 4, then there are two conjugacy classes of elements in τ0 but each element in τ0 is of order q. Therefore; 1 q − 1 q − 1 Z = (tq + .mon(x ) + .mon(x ) (G,X) |G| 1 2 1 2 2 +q(summation of all monomials of the nontrivial elements in cyclic

subgroups C q−1 )), 2 where x1 and x2 are representatives of elements in the ﬁrst and the second conjugacy classes respectively but since all τ0 are of order p then they will have the same monomial and thus Z(G,X) can be written as;

1 Z = (tq + (q − 1).monomial of an element in τ (G,X) |G| 1 0 +q(summation of all monomials of the nontrivial elements in cyclic

subgroups C q−1 )) 2 That is,   1 X Z = tq + (q − 1) .mon(x) + q mon(g) (5) (G,X) |G|  1  g∈Cq−1\{I} where x ∈ τ0. Let x ∈ τ0, then π (x) = 0 It follows from Theorem 2.4 that, π (xp) = q and if l < p, π xl = 0 Now if 0 < l < p then,

1 X l α = π x i µ (i) l l i|l 1 X = 0µ (i) = 0 l i|l 312 Geoﬀrey Ngovi Muthoka, Ireri Kamuti and Mutie Kavila and

1 X p α = π x i µ (i) p p i|p 1 = [π (xp) − π (x)] p 1 q = [q − 0] = . p p Therefore the resulting monomial is;

q p tp (6)

If g ∈ τ1, then π (g) = 1, π gd = q where d is the order of g, π gl = 1 when l < d

1 X l α = π g i µ (i) l l i|l 1 X = (1)µ (i) l i|l 1 X = µ (i) = 0 l i|l

1 X d α = π g i µ (i) d d i|d   1 d X d = π g µ(1) + π g i µ (i) − π (g) d   16=i|d 1 = π gd − π (g) d 1 q − 1 = [q − 1] = d d Thus the resulting monomial is;

q−1 d t1td (7)

Substituting for mon(x) (in 6) and mon(g) (in 7) in Equation 5 we get; q q−1 1 q p P d Z(G,X) = t1 + (q − 1) tp + q q−1 ∅(d)t1t |G| 16=d| 2 d The cycle index of the aﬃne square(q) group 313

Example 3.1.1 Let p = 5, r = 2 ⇒ q = 52,X = GF (25) and |G| = 300. Possible values of d are; 2, 3,4, 6 and 12. Then ∅ (2) = 1, ∅ (3) = 2, ∅ (4) = 2, ∅ (6) = 2 and ∅ (12) = 4. Substituting in Theorem 3.3, we have;

1 25 5 12 8 6 4 2 Z(Aff (25),X) = t1 + 24t5 + 25t1t2 + 50t1t3 + 50t1t4 + 50t1t6 + 100t1t12 . 300

3.2 Expressing the cycle index of the Aﬃne square(q) group in terms of the cycle index of the elementary abelian group Pq and the cyclic group C q−1 2

The equation in Theorem 3.3 can be simpliﬁed as;

  1 q X q−1 Z = tq + (q − 1) t p + q ∅(d)t t d (G,X) |G|  1 p 1 d  q−1 16=d| 2   q q−1 1 q p 1 q X d 1 q = q−1 t1 + (q − 1) tp + q−1 qt1 + q ∅(d)t1td  − q−1 t1 q 2 q 2 q−1 2 16=d| 2   q−1 1 1 q X d 1 q = q−1 Z(Pq,X) + q−1 t1 + ∅(d)t1td  − q−1 t1 2 2 q−1 2 16=d| 2 1 1 q = q−1 Z(Pq,X) + Z(C q−1 ,X) − q−1 t1 2 2 2 1 1 = Z(Pq,X) + Z(C q−1 ,X) − Z(1,X). (8) 2 C q−P C q−1 2 2

Example 3.2.1 Let p = 3, r = 2 ⇒ q = 32,X = GF (9) and |G| = 36 and 8 1 9 3 P d Z(G,X) = 36 t1 + 8t3 + 9 16=d|4 ∅(d)t1td (from Theorem 3.3) 314 Geoﬀrey Ngovi Muthoka, Ireri Kamuti and Mutie Kavila which can be simpliﬁed as;   1 1 X 7 1 Z = t9 + 8t3 + 9t9 + 9 ∅(d)t t d − t9 (G,X) 9 (4) 1 3 9 (4)  1 1 d  4 1 16=d|7 1 1 = Z + Z − t9 4 (P9,X) (C4,X) 4 1 1 1 = Z + Z − Z (from Equation 8). 4 (P9,X) (C4,X) 4 (1,X)

Acknowledgements. Supported by National Research Fund; NRF

References

[1] Harald, F., Cycle indices of linear, aﬃne and projective groups. Linear Algebra and Its Applications, 263 (1997), 133–156. https://doi.org/10.1016/s0024-3795(96)00530-7

[2] Harrison, M. A. and High, R. G., Cycle index of a product of permutation groups. Combinatorial Theory, 4 (1968), 277-299. https://doi.org/10.1016/s0021-9800(68)80008-0

[3] Jason, F., Cycle indices for the ﬁnite classical groups, Journal of Group Theory, 2 (1999), 251-289.

[4] Kamuti, I. N., Combinatorial formulas, invariants and structures associated with primitive permutation representations of PGL(2,q) and PSL(2,q), PhD Thesis, Mathematical Studies Southampton University, UK, 1992.

[5] Kangogo M. R., Ranks and subdegrees of the cyclic group, the dihedral group and the aﬃne group and associated suborbital graphs, PhD Thesis, Pure Mathematical Kenyatta University, Kenya, 2015.

[6] Peter, J. and Jason, S., The cycle polynomial of permutation group. The Electronic Journal of Combinatorics, 25 (2017), 1-16.

[7] P´olya, G., Kombinatorische anzahlbestimmungen f¨urgruppen, graphen und chemische verbindungen, Acta Mathematics, 68 (1937), 145-254. https://doi.org/10.1007/bf02546665

Received: August 3, 2019; Published: September 3, 2019