Journal of Computer and System Sciences 1398 journal of computer and system sciences 53, 19 (1996) article no. 0045 An O(n) Algorithm for Abelian p-Group Isomorphism and an O(n log n) Algorithm for Abelian Group Isomorphism Narayan Vikas* Department of Computer Science and Engineering, Indian Institute of Technology (IIT), Delhi, Hauz Khas, New Delhi - 110016, India Received October 6, 1988; revised July 20, 1993 and October 9, 1995 and the order of G is |G|. We will use ab to denote a V b. For The isomorphism problem for groups is to determine whether two a # G, the order of a, denoted as o(a), is the smallest integer finite groups are isomorphic. The groups are assumed to be represented ie1 such that ai=e (ai means a is multiplied by itself i by their multiplication tables. Tarjan has shown that this problem can be times). Further, if G is finite then o(a)||G| (to read as o(a) done in time O(nlog n + O(1)) for groups of cardinality n. Savage has claimed an algorithm for showing that if the groups are Abelian, is a factor of |G|). A finite group G is said to be an Abelian m isomorphism can be determined in time O(n2). We improve this bound p-group if G is Abelian and |G|=p , where p is a prime and give an O(n log n) algorithm for Abelian group isomorphism. number and me0 an integer. Thus, every element in an Further, we show that if the groups are Abelian p-groups then Abelian p-group has order p power. If G is a finite Abelian isomorphism can be determined in time O(n). We also show that the group and p ||G|, where p is a prime number, then there is elementary divisor sequence for an Abelian group can be determined in time O(n log n) and for an Abelian p-group it can be determined in time an element a # G such that o(a)=p. If a group G has an ele- k i O(n). ] 1996 Academic Press, Inc. ment of order p then G has an element of order p , for all i<k, where p is a prime number and i, ke0 are integers. An element x of a group with o(x)appower, say, o(x)=pj, 1. DEFINITIONS AND BACKGROUND p a prime number and je0 an integer, will be called a prime power order element or just a p power order element. Two groups G1 and G2 are said to be isomorphic, written Before we give our algorithms, we state some relevant as G1rG2 , if there is a function f: G1 Ä G2 which is one-to- definitions and results of group theory. Readers may refer to one and onto such that for all a, b # G1 , f(ab)=f(a) f(b). [3, 5, 6] for a study on group theory. Notice that the product ab is computed in G1 and the A nonempty set G is said to form a group if there is in G product f(a) f(b) is computed in G2 . G1 r3 G2 will denote such that a binary operation *, called the product, defined that G1 and G2 are not isomorphic. The direct product of two groups G and G is the group (i) a, b # G implies that a V b # G (closed); 1 2 G1 _G2=[(a, b)|a#G1,b#G2] whose multiplication is (ii) a, b, c # G implies that a V (b V c)=(a V b) V c defined by (a1 , b1) V (a2 , b2)=(a1a2, b1b2). The product (associative law); a1 a2 is computed in G1 and the product b1 b2 is computed in (iii) there exists an element e # G such that a V e= G2 . The identity of G1 _G2 is (e1 , e2), where e1 and e2 e V a=a for all a # G (the existence of an identity element are the identities of G1 and G2 respectively. Note that in G); |G1 _G2 |is|G1| times |G2 | and o(a, b)=lcm(o(a), o(b)), \a # G , \b # G , where o(a) is the order of the element a in (iv) for every a # G there exists an element a&1 # G 1 2 G , o(b) is the order of b in G , and lcm is the least common such that a V a&1=a&1 V a=e (the existence of inverses 1 2 multiple operator. in G). Let a be an element of a group G such that o(a)=k. The A group G is said to be Abelian (or commutative), after set (a)=[a, a2, a3, ..., ak&1, ak=e] is a group (subgroup the name of the famous mathematician Abel, if for every of G) called the cyclic group generated by a. One can easily i a, b # G, a V b=b V a. G is said to be a finite group if it has prove that o(a )=lcm(i, o(a))Âi, \i=1, 2, ..., k. Note that finite number of elements. |G| denotes the cardinality of G two finite cyclic groups (a) and (b) are isomorphic iff o(a)=o(b). It is easy to see that for two finite groups to be isomorphic it is a necessary condition that the orders of the * Current address: School of Computing Science, Simon Fraser Univer- groups be the same. In case of finite cyclic groups it is also sity, Burnaby, British Columbia, Canada V5A 1S6. a sufficient condition for isomorphism. Let C(k) represent 1 0022-0000Â96 18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. File: 571J 139801 . By:BV . Date:29:08:96 . Time:11:47 LOP8M. V8.0. Page 01:01 Codes: 6426 Signs: 4489 . Length: 60 pic 11 pts, 257 mm 2 NARAYAN VIKAS a cyclic group of order k. Any Abelian p-group is Abelian groups, it is sufficient to find just the orders of the isomorphic to a direct product of cyclic groups of p power prime power order elements in the groups and we give an orders. Further, for any Abelian p-group G of order O(n log n) algorithm in Section 3 for finding orders of all the pm, me1, G can be represented as prime power order elements in an Abelian group and an O(n) algorithm in Section 2 for finding orders of all the GrC( pr1)_C( pr2)_}}}_C(prk), (I) element in an Abelian p-group. However, if we were inter- k where i=1 ri=m, ri eri+1>0, 1Ei<k. ested in finding the EDS as well, then we find it still in We will call the sequence pr1, pr2, ..., prk the elementary O(n log n) time for an Abelian group and in O(n) time for divisor sequence (EDS) for G.If|G|=1, the EDS of G is an Abelian p-group for which the corresponding algorithms empty. Two Abelian p-groups are isomorphic iff they have are given in Section 4. Section 5 gives some future directions. the same EDS. An appendix is also attached giving detailed forms of our For any finite Abelian group G* of order n>1, G* can be algorithms with documentations. represented as 2. ABELIAN p-GROUP ISOMORPHISM G*rG( pa1)_G( pa2)_ } } } _G( pas), (II) 1 2 s Let us consider the representation of an Abelian p-group m where 1<p1<p2<}}}<ps,pi a prime number, and G as described in (I) in Section 1, where |G|=p and there ai>0 an integer \i=1, 2, ..., s such that n= is an ordering on ri 's (ri 's are in nonincreasing order). We a1 a2 as ai ai p1 p2 }}}ps ,G(pi ) is an Abelian pi -group of order p i would like to obtain a formula for the number of elements ai t r1 r2 rk and the elements of G( pi ) are all the elements of G* whose of order p in C( p )_C( p )_ } } } _C( p ). Since there is ai orders are pi power, \i=1, 2, ..., s. Note that each G( pi ) exactly one element of order 1, we will assume t>0. In order can be further expanded to a direct product of cyclic groups to obtain this result we first find the number of elements of t ri of pi power orders as explained above in (I). Thus, G* order p in C( p ), \i=1, 2, ..., k. Note that if ri<t then will be isomorphic to a direct product of cyclic groups of C( pri) does not contain any element of order pt. So we con- prime power orders and the EDS of G* will be EDS(1) & centrate on those cyclic groups whose orders are at least pt. EDS(2) & }}}&EDS(s), where EDS(i) denotes the EDS It is easy to see that if (x) is a cyclic p-group having an ai t of G( pi ), \i=1, 2, ..., s and & is a concatenation operator element y with o( y)=p then (y) is a cyclic subgroup of (for sequences s1 and s2 , s1 & s2 is a sequence with s1 and s2 (x) and (y) contains all the elements of (x) that are of t t its subsequences such that s1 & s2 is the sequence s1 followed order Ep . (Thus the number of elements of order Ep in t i t i t by s2).