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Internat. J. Math. Math. Sci.Vol. 2 # (1979) 487-491

487

EQUIVALENCE CLASSES OF MATRICES OVER A FINITE FIELD

GARY L. MULLENDepartment of Mathematics

The Pennsylvania State UniversitySharon, Pennsylvania 16146

(Received October 26, 1978)

ABSTRACT. Let Fq --GF(q) denote the finite field of order q and F(m,q) the

ring of mxm matrices over Fq. Let be a group of permutations of Fq. If

A B F(m,q) then A is equivalent to B relative to if there exists

such that (A) B where (A) is computed by substitution. Formulas are

given for the number of equivalence classes of a given order and for the total

number of classes induced by a cyclic group of permutations.

KEY WORD AND PHRASES. Equivalence, prmutation, automorphm, finite field.

198 MATHEMATICS SUBJECT CLASSIFICATION CODES. Primary 15A33; Secondary 15A24.

i. INTRODUCTION.

In the series of papers [2], [4], [8], and [9], L. Carlitz, S. Cavior, and

the author studied several forms of equivalence of functions over a finite field.

In this paper we study a form of equivalence defined on the ring of mxm matrices

over a finite field.

488 G.L. MULLEN

Let F GF(q) denote the finite field of order q and F(m,q) the ringq

of mxm matrices over Fq. Let be a group of permutations on Fq so that

is isomorphic to a subgroup of S the symmetric group on q letters.q

2. GENERAL THEORY.

A permutation of F is a function from F onto F and so by theq q q

Lagrange Interpolation Formula [6,p.55], is expressible as a polynomial of degree

xn+ .+ alx + a where each ai

e F andless than q. Suppose (x) an o q

n <_ q-l. Then by substitution we may compute the mxm matrix (A)

naAn +’’’+ alA + aol where I is the mxm identity matrix. If is a groupm m

of permutations then we can make

DEFINITION i. If A,B e F(m,q) then A is equivalent to B relative to

if B (A) for some

As this relation is an equivalence relation on F(m,q) we let (A,) denote

the order of the class of A relative to and () denote the number of

classes induced by .DEFINITION 2. If A E F(m,q) then a permutation E is an automorphism

of A relative to if (A) A.

Let Aut(A,R) and u(A,R) denote the group and number of automorphisms of

the matrix A relative to R. If #(A) B for some then

Aut(B,R) #Aut (A,R) -I (2.1)

so that U(B,) u(A,). Thus the number of automorphlsms depends only upon

the class and not on the particular matrices within the class. One can now

easily prove

THEOREM 2i. Let A F(m,q). Then for any group

(A,) (A,)--II (2.2)

where II denotes the order of the group .If is a permutation let l(,m) denote the number of mxm matrices A

MATRICES OVER A FINITE FIELD 489

such that (A) 0. Suppose (x) anxn +’"+ alx + ao where each

an + o. Then if (A) A we have the matrix equation

An + an-! __An-1 + +() A +._.ao Im 0.an an an

Consider the monic polynomial

E(x) xn + an-I xn-I + + (ai-l) x + aoa a ann n

over GF(q). Hodges in [7] gives a formula for the number N(E,m) of mxm

matrices A over GF(q) such that E(A) 0. Clearly l(,m) N(E,m) and

moreover this number can be computed as follows.

e F andai q

(2.3)

Berlekamp in [i] has given an algorithm for factoring polynomials over Fq

in terms of solving a system of linear equations over F Suppose E hasq

h hthe prime factorlzation E PII PsS where the Pi’s are distinct monic

polynomials, hi >_ I and the degree of Pi is d

ifor i l,...,s. Let g(t,d)

represent the number of non-singular matrices of order t > i over GF(qd) so

that, as is well known, g(t,d) tli=o(qdt qdi/. We also set g(o,d) i.

Let be a partition of m defined by

s him i=II d

i JZ--1 Jkij kij _> o. (2.5)

For i l,...,s let

hi hi

bi(2 (u-l) + 2Ukiu E

u--i[kiu

v--u+l kivs

and set a(r) Z dibi(r). Then as shown by I-lodges in [7].i--i

s hil(,m) N(E,m) g(m,l) Z g-a()i=l

J4=IH g(kij,di)-i

(2.6)

(2.7)

where the sum is over all partitions of m given by (2.5).

3. CYCLIC GROUPS.

Let be a cyclic group of order n, say <>, and let H(t) denote

490 G.L. MULLEN

the subgroup of 0 of order t, where tln. It follows that H(t) <nl t>.Let M(t,m) denote the number of mxm matrices A such that Aut(A,) H(t).

THEOREM 3.1. For each divisor t of n

,m) z N(E m)M(t,m) N(E#n/t n/uwhere the sum is over all u for which u ln, t[u, and t u.

PROOF. The number of mxm matrices A such that H(t) <_ Aut(A,) is given

by N(E m). From this we subtract those for which the containment ifn/

proper.

COROLLARY 3.2. For all tln there are tM(t,m)/n classes of order n/.t

and

l(0) 1 l tM(t,m). (3.2)n tln

Following Definition 3.1 of [8] we say that two groups 01 and 2 induce

equivalent decompositions of F(m,q) is they induce the same number of classes

of the same size. It is easy to see that if < I > and 2 < 2 > are

cyclic groups of order n then I and 2 will induce equivalent decompositions

of F(m,q) if for each t ln #?/t and /t have the same number of distinct

prime factors of the same degree and multiplicity.

As a simple illustration of the above theory let m 2 and consider

(x) x3 over GF(5) so that if <#> then II 2. We have

E#(x) x3 x x(x-l)(x+l) so that s 3, PI x, P2 x-l, P3 x+l and

hi

di

i for i 1,2,3. There are six partitions defined by (2.5) which

are given by 2 kll + k21 + k31 where o <_ klj <_ 2. Clearly bi() o for

i 1,2,3 so that a() o for each partition . One checks that g(2,1)

(q2 l)(q2 q) 480, g(l,l) q-i 4 and g(o,l) i. Hence using (2.7)

it is not difficult to check that N(E,2) 93 so that by (3.1), M(2,2) 93

and thus M(I,2) 532. Hence by Corollary 3.2 there are 93 classes of order

one and 266 classes of order two so that by (3.2) A() 359.

MATRICES OVER A FINITE FIELD 491

REFERENCES

i. Berlekamp, E. R., "Factoring polynomials over finite fields," Bell SystemTechnical J. 46 (1967), 1853-1859.

2. Carlitz, L., "Invariantive theory of equations in a finite field," Trans.Amer. Math. Soc. 75 (1953), 405-427.

3. Carlitz, L., "Invariant theory of systems of equations in a finite field,"J. Analyse Math. 3__ (1953-54), 382-413.

4. Cavior, S. R., "Equivalence classes of functions over a finite field,"Acta Arith. i0 (1964), 119-136.

5. Cavior, S. R., "Equivalence classes of sets of polynomials over a finitefield," Journ. Reine Anew. Math. 225 (1967), 191-202.

6. Dickson, L. E., Linear Groups with an Exposition of the Galois Field Theory,Dover Publications, Inc., New York, 1958.

Hodges, J. H., "Scalar polynomials equations for matrices over a finitefield," Duke Math. J. 25 (1958), 291-296.

8. Mullen, G. L., "Equivalence classes of functions over a finite field,"Acta Arith. 29 (1976), 353-358.

9. Mullen, G. L., "Equivalence classes of polynomials over finite fields,"Acta Arith. 31 (1976), 113-123.

i0. Mullen, G. L., "Equivalence classes of sets of polynomials over finite

fields," To appear, Bull. Calcutta Math. Soc.

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