Automorphism Classes of Elements in Finitely Generated Abelian Groups
aa r X i v : . [ m a t h . G R ] S e p AUTOMORPHISM CLASSES OF ELEMENTS IN FINITELYGENERATED ABELIAN GROUPS.
CHARLES F. ROCCA JR.
Abstract.
We will show that every element of a finitely generated abeliangroup is automorphically equivalent what we will define to be a representativeelement in a repeat-free subgroup , and for finite abelian groups we can countthe number of automorphism classes of elements. Representative Elements
A finite abelian p -group G p can be written G p = Z k p r ⊕ · · · ⊕ Z k n p rn , r i < r i +1 f or all i. However for this paper we will write such a group as follows G p = [ Z p r ⊕ · · · ⊕ Z p rn ] ⊕ [ Z k − p r ⊕ · · · ⊕ Z k n − p rn ] , r i < r i +1 . This splits the group into two subgroups, a repeat-free subgroup , G rfp = Z p r ⊕ · · · ⊕ Z p rn , which contains one copy of each of the factors Z p ri and a remainder subgroup , G rmp = Z k − p ri ⊕ · · · ⊕ Z k n − p rn , that contains the remaining factors of the group.For the remainder of this article let G be a finitely generated abelian group ofthe form G = G p ⊕ · · · ⊕ G p m ⊕ Z l , where l and m ≥ p i are distinct primes with p i < p i +1 for all i . Theabove definition of a repeat free subgroup can be extended to any finitely generatedabelian group G as follows. Definition 1.
Given a finitely generated abelian group G a repeat-free subgroup has one of the following forms (1) G rf = G rfp ⊕ · · · ⊕ G rfp m ⊕ Z if l and m ≥ , or (2) G rf = G rfp ⊕ · · · ⊕ G rfp m if l = 0 and m ≥ , or (3) G rf = Z if l ≥ and m = 0 . The next definition defines our set of representative elements in a repeat-free finiteabelian p -group. Definition 2.
An element g = ( g , . . . , g n ) of a repeat-free finite abelian p -group G rfp = Z p r ⊕ · · · ⊕ Z p rn r i < r i +1 for all i is a representative element if (1) for all i g i is either or a power of p , Mathematics Subject Classification.
Primary 20K30, 20K01. (2) for all i, j if i < j , then g i < g j , and (3) for all i, j if i < j , then the order of g i in Z p ri is less than the order of g j in Z p rj . We will show that in a finite abelian p -group every element is automorphicallyequivalent to a representative element of a repeat-free subgroup. Therefore welet a representative element of a finite abelian p -group be an element of a repeat-free subgroup which is a representative element in that subgroup. Extending thisdefinition to all finitely generated abelian groups we get. Definition 3.
Let G ba a finitely generated abelian group, then we say g =( g , g , . . . , g m , Z ) ∈ G is a representative element if (1) for all j , g j is a representative element of G p j , (2) the element Z ∈ Z l is of the form ( z, , . . . , and (3) for all j , each term of g j is relatively prime to z . So, the primary goal of this paper is the following theorem.
Theorem 1.
Every element of a finitely generated abelian group is automorphi-cally equivalent to an unique representative element.
The proof of this result has two main steps:(1) Every element is automorphically equivalent to an element in a repeat-freesubgroup.(2) Every element of a repeat-free subgroup is automorphically equivalent to arepresentative element.The following lemma, in which we think to endomorphisms of finite abelian groupsas matrices, is a major tool in proving these results.
Lemma 1. [1]
An endomorphism ϕ of G p = Z p r ⊕ Z p r ⊕ · · · ⊕ Z p rm , p primeand r i ≤ r i +1 for all i , is an automorphism if and only if the reduction of ϕ modulo p , written ϕ p , is an automorphism of Z mp . The following proposition establishes the first step in the proof of the theorem.
Proposition 1.
Every element in a finitely generated abelian group G is auto-morphically equivalent to an element in the repeat-free subgroup. In particular: a) Each element g = ( g , . . . , g n ) ∈ Z np r , n ≥ is automorphically equivalentto an element of the form ( p l , , · · · , , where the order of g is p r − l . b) Each element g = ( g , . . . , g k ) of Z k is automorphically equivalent to oneof the form ( d, , · · · , , where d = gcd( g , ..., g k ) .Proof: Let g ∈ Z np r ; each term in g is of the form g i = a i p l i , where a i is rela-tively prime to p. Therefore g is mapped automorphically to the element g ′ =( p l , · · · , p l n ) by multiplying each g i by the inverse of a i modulo p r .The order of g ′ which is the same as the order of g , is p r − l . Therefore at leastone term of g ′ is equal to p l , without loss of generality let g ′ = p l . Also, every termof g ′ will be divisible by p l , that is l i ≥ l for all i . Therefore · · · − p l − l · · · − p l n − l · · · p l p l ... p l n = p l , UTOMORPHISM CLASSES OF ELEMENTS IN FINITELY GENERATED ABELIAN GROUPS.3 where it is clear from the lemma 1 that this matrix is an automorphism, thusproving part (a).The element g = ( g , . . . , g k ) of Z k can be reduced using the Euclidian Algorithm bythe repeated application of elementary row operations and this establishes (b). (cid:3) To show that every element of a repeat-free subgroup is automorphically equiv-alent to a representative element we first restrict ourselves to a finite repeat-free p -group and define a basic reduction of an element. Definition 4.
Given a repeat-free p -group G rfp = Z p r ⊕ · · · ⊕ Z p rn , r i < r i +1 f or all i and an element g = ( p l , . . . , p l n ) , a Basic Reduction about position i of g is thefollowing:for every j = i with p l j ≥ p l i and p r j − l j ≤ p r i − l i replace p l j with 0,i.e. if the j th term has a greater value but lesser order than the i th term replace itby 0.If the number of non-zero terms decreases, then the basic reduction is a non-trivial basic reduction , otherwise it is a trivial reduction . We will say that anelement is reduced if there are no non-trivial basic reductions. The matrix for the reduction transformation has elements a lk = if l = k − p l j − l i if l = j and k = i otherwise which is similar to the matrix in Proposition 1 and is again by by Lemma 1 is anautomorphism. Lemma 2.
Let G rfp = Z p r ⊕ · · · ⊕ Z p rn , r i < r i +1 f or all i be a repeat-free p -group. Then an element g ∈ G rfp is reduced if and only if g is arepresentative element.Proof: Let g = ( p l , . . . , p l n ) be a reduced element of G rfp and let 1 ≤ i < j ≤ n . Case 1: If p l i ≥ p l j , then the order of p l i in Z p ri is strictly less than theorder of p l j in Z p rj . Therefore, we can perform a basic reduction about position j ,contradicting the assumption that g is reduced. Case 2: If p l i < p l j and the order of p l i in Z p ri is greater than or equal to theorder of p l j in Z p rj , then we can perform a basic reduction about position i againcontradicting the assumption that g was reduced. Therefore, if g in G rfp is reduced,then it is a representative element.Now suppose that g is a representative element. Let 1 ≤ i < j ≤ n so that p l i < p l j and the order of p l i in Z p ri is strictly less than the order of p l j in Z p rj ,when p l i and p l j are non-zero. If we performed a basic reduction about position i ,then p l j would be unchanged since the order of p l i is less than that of p l j . Similarly,if we performed a basic reduction about position j , p l i would remain unchangedsince p l i < p l j . Since i and j are arbitrary if g is a representative element, then g is reduced. (cid:3) We are now in a position to prove the main result for the case of repeat-free p -groups. CHARLES F. ROCCA JR.
Proof: (Theorem 1 for repeat-free p -groups) Let g = ( g , . . . , g n ) be an elementin a finite repeat-free p -group G rfp . Applying Proposition 1 we know that g isautomorphically equivalent to some g ′ in which each term is a power of p . Now,if g ′ is not a reduced element, then we can perform a non-trivial basic reductionand the the resulting element, which is automorphically equivalent to g ′ , will havestrictly fewer non-zero terms. Since the rank of G rfp is finite the process of makingnon-trivial reductions will terminate, and the resulting element will be a reducedelement and therefore a representative element.Finally we show that representative elements of automorphism classes are unique. Claim 1. If g and h are automorphically equivalent representative elements in G p , then g = h .Proof: (of claim) Let g = ( g , . . . , g n ), h = ( h , . . . , h n ) and let A be the matrixrepresenting the automorphism taking h to g . Suppose h i = 0 and let p l i , 0 ≤ l i ≤ r i be the order of h i in Z p ri so that we may write h i = p r i − l i . By Lemma 1 theautomorphism A must have the form A = a a · · · a n p r − r a a · · · a n ... ... p r n − r a n p r n − r a n · · · a nn where a ii is relatively prime to p . Therefore, assuming that h i = 0, g i = p r i − l a i + · · · + p r i − l i a ii + · · · + p r n − l n a in . Since h is a representative element, for any non-zero term h j if j < i , then the orderof h j in Z p rj is less than the order of h i in Z p ri ; i.e. l j < l i . Hence, r i − l j > r i − l i and p r i − l i divides p r i − l j . If j > i , then h i < h j and p r i − l i divides p r j − l j . Thus g i = p r i − l i ( p l i − l a i + · · · + a ii + · · · + p r n − l n − ( r i − l i ) a ni ) . Since a ii is relatively prime to pb i = ( p l i − l a i + · · · + a ii + · · · + p r n − l n − ( r i − l i ) a ni ) . is also relatively prime to p . Thus, when h i is non-zero, g i = b i h i , b i relativelyprime to p . However g is a representative element therefore g i is a power of p , b i = 1 and g i = h i .An identical argument shows that h i = g i when g i is non-zero. Therefore for all i , g i = h i and so g = h . (cid:3) If G p is any finite abelian p -group, then every element of G p is automorphicallyequivalent to a unique representative element in G rfp , which is also a representativeelement for G p . Finally, if G is any finitely generated abelian group, then wehave established that any element of G is automorphically equivalent to elementsatisfying the first two conditions of the definition of representative element. It canbe shown that any element satisfying the first two conditions is automorphicallyequivalent to one that satisfies the third condition and has the same number orfewer nonzero terms. As an illustration consider the following example.Let G rf = Z p r ⊕ Z p r ⊕ Z and let g = [ p l , p l , kp ] be any element of G rf satisfying the first two conditions of the definition of a representative element but UTOMORPHISM CLASSES OF ELEMENTS IN FINITELY GENERATED ABELIAN GROUPS.5 not the third. Then p r − − p l − p r − − p l − p l p l kp = kp and so g is automorphically equivalent to an element satisfying the third conditionof our definition and has strictly fewer nonzero terms.2. Counting Automorphism Classes
Having completed the main result we will now show how to count the numberof representative elements in a given finite abelian group.Let G rfp = Z p r ⊕ Z p r ⊕ · · · ⊕ Z p rk where r i < r i +1 for all i be a repeat-free p -group and g = ( p l , p l , . . . , p l k ) be an automorphism class representative in G rfp . The non-zero terms of g are both increasing and order increasing, thus for0 ≤ i < j ≤ k if p l i and p l j are non-zero terms of g we know that p l i < p l j and p r i − l i < p r j − l j . Hence, l i < l j < l i + ( r j − r i ) . We shall refer to the value r j − r i as the gap between the terms Z p ri and Z p rj of G rfp , and in particular we will denote r i +1 − r i by n i . As an immediate consequenceof the above conclusion we know that if p l i is non-zero, then either p l j is zero or wehave at most r j − r i − l j . Further, if j = i + 1 and n i = r i +1 − r i = 1,then one or the other of these two terms must be zero. Therefore, if the gap betweentwo terms of a group is one, then in every automorphism class representative atleast one of the terms will be zero.For motivation let us count the number of automorphism class representativesin G rfp = Z p r ⊕ Z p r ⊕ Z p r where r < r < r . We begin by counting the numberof automorphism classes with a given number of non-zero terms. r + r + r r ( n −
1) + r ( r − r −
1) + r ( n − r ( n − n − r = r + n and r = r + n + n therefore the above equations can berewritten as, r + ( r + n ) + ( r + n + n )2 r ( n −
1) + r ( n + n −
1) + ( r + n )( n − r ( n − n − CHARLES F. ROCCA JR.
Non-Zero Terms r + 2 n + n r (2 n + 2 n −
3) + n n − n r ( n n − n − n + 1)and the number of automorphism classes of elements equals r (1 + n + n + n n ) + 1 + n + n + n n = ( r + 1)( n + 1)( n + 1) . Proposition 2.
The number of automorphism classes of elements in a repeat-freefinite abelian p -group G rfp = Z p r ⊕ Z p r ⊕ · · · ⊕ Z p rk , r i < r i +1 f or all i is equal to ( r + 1)( n + 1)( n + 1) · · · ( n k − + 1) , where n i = r i +1 − r i . Proof:
We have shown above that this is true for a group with three terms. Weproceed by induction on the number of terms in G rfp . By induction the group H rfp = Z p r ⊕ Z p r ⊕ · · · ⊕ Z p rk − has ( r + 1)( n + 1)( n + 1) · · · ( n k − + 1) automorphism classes of elements. Thisis also the number of automorphism class representatives in G rfp in which the k th term is 0. Therefore we complete the proposition with the following claim. Claim 2. If < j ≤ k , then in G rfp = Z p r ⊕ Z p r ⊕ · · · ⊕ Z p rk the number of automorphism class representatives in which the j th term is the lastnon-zero term is ( r + 1)( n + 1)( n + 1) · · · ( n j − + 1) n j − . Proof: (of claim) If j = 2, then there are r automorphism class representatives inwhich the second term is the only non-zero term and r ( n −
1) in which the first andsecond terms are the only non-zero terms. Therefore the number of automorphismclass representatives in which the second term is the last non-zero term is r ( n −
1) + r = r n − r + r + n = ( r + 1) n . Now let 2 < j ≤ k , if 2 ≤ i < j and the i th term is the last non-zero term of anelement prior to the j th term, then there are ( r j − r i −
1) non-zero choices for thevalue of the j th term and by induction( r + 1)( n + 1)( n + 1) · · · ( n i − + 1) n i − ( r j − r i − j th term is the last non-zero termand the i th term is the second to last non-zero term. So letting A i = ( r + 1)( n + 1)( n + 1) · · · ( n i − + 1) n i − and summing over all i between 2 and j , the total number of automorphism classesin which the j th term is the last non-zero term is A j = r j + r ( r j − r −
1) + j − X i =2 A i ( r j − r i − . However for all 1 ≤ i < j ≤ k : r j = r j − + n i − and ( r j − r i −
1) = ( r j − − r i − n j − )so we may rewrite the previous equation as( r j − + n j − ) + ( r ( r j − − r −
1) + r n j − ) + j − X i =2 ( A i ( r j − − r i −
1) + A i n j − )And since A j − i = r j − + r ( r j − − r −
1) + j − X i =2 A i ( r j − − r i − A j = A j − + n j − + r n j − + A j − ( r j − − r j − −
1) + j − X i =2 A i n j − = A j − − A j − + r n j − + n j − + j − X i =2 A i n j − ! = r n j − + n j − + j − X i =2 A i n j − = n j − ( r + 1) + j − X i =2 A i ! = ( r + 1)( n + 1)( n + 1) · · · ( n j − + 1) n j − , thus proving the claim. In order to finish the proposition we observe that, byinduction, the number of automorphism classes in which the k th term is zero is( r + 1)( n + 1)( n + 1) · · · ( n k − + 1)and from the claim the number of automorphism classes in which the k th term isnon-zero is ( r + 1)( n + 1)( n + 1) · · · ( n k − + 1) n k − , and the sum of these two gives the desired result. (cid:3) Since automorphisms respect the prime decomposition of finite abelian groupswe get this final general result.
Theorem 2.
The number of automorphism classes of elements in a finite abeliangroup G = G p ⊕ · · · ⊕ G p m CHARLES F. ROCCA JR. where G p j = Z k p r j ⊕ · · · ⊕ Z k n p rnj , r i < r i +1 is equal to the product of the number of automorphism classes in each individual G rfp j . References [1] C.F. Rocca Jr. and E.C. Turner,
Test Elements in Finitely Generated Abelian Groups , Int.J. Algebra and Computation, Vol. 12, No. 4 (2002) pp.569-573.[2] J.C. O’Neill and E. C. Turner,
Test elements in direct products , Int. J. Algebra and Comput,Vol. 10, No. 6 (2000), pp.751-756.
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