Vol. 5 No 10 - Pi Mu Epsilon

~A NESTED PRIME NUMBERMAGIC SQUAREReverend Victor Feser of St. Ambrose Church, St. Louis, has pointedout an example of a 13 x 13 nested magic square consisting entirely ofprime numbers.^-(See in this connection the article "1-lagic SquaresWithin Magic Squares" by Joseph Moser, this Journal, 5, No. 8 (Spring1973), p. 430.) Each smaller square centered at 5437 is also a magic square,lbom the Recreational Mathematics Journal, No. 5 (October 1961), p. 28.REFEREES FOR THIS ISSUEThe editorial staff sends a note of appreciation to the followingpersons who freely gave of their time to evaluate papers submitted forpublication prior to this issue : Joseph Lehner, University of P-i rtsburgh ;Walter Growney, Susquehanna University; and members of the mathematicsdepartment at the University of Oklahoma, R. V. Andree, Patrick Cassady,Bradford Grain, John Green, Andy Ilagid, Lee Pound (computer science 1,W. 1'. Reid, and Kirby Smith.The Journal also acknowleges with gratitude the typist For thisissue, Theresa Killgore, who is a senior at the University of 'Jklahoma.NICENESS OF THE SOCLE AND A CHARACTERIZATIONOF GROUPS OF BOUNDED ORDERby S. U. TcLU.ey1UutVin Kentnckq Univm'LtqA classification result (that tells when or how two algebraic sys-tems are of the same kind) is one of the most desired results in thestudy of any algebraic system.theorems is that of Ulm [lo],One of the most notable classificationwhich classifies countable reduced abelianprimary groups in terms of a set of numerical invariants.This classi-fication using the same invariants has been extended to a much largerclass of abelian p-groups, the class of totally projective groups, inthe work of Nunke [g], Hill [5] (see Griffith [3]), and Crawley andHales [l].Hill introduced the concept of "nice subgroup" and established thecharacterization of a totally projective group as a reduced p-group thatcontains "enough" nice subgroups.Moreover, Hill was able to show thatthe class of totally projective groups is the largest reasonable classof reduced primary abelian groups that can be classified by their Ulminvariants.~f G is an abelian p-group (p prime) and n is a non-negative integer,then G [ ~ ] denotes the subgroup of G consisting of all elements havingorder less than or equal to pn; ~[p] is called the socle of G. In thisnpaper a necessary and sufficient condition is found for G[p ] to be anice subgroup of the reduced abelian p-group G (see Theorem 1). Thiscondition, together with the result of Hill [4], leads to the character-ization of a reduced bounded p-group as a reduced p-group in which allsubgroups are nice (see Theorem 4).Pfie^Mn.inCm.&'iAll groups in this paper are assumed to be additively written re-duced abelian p-groups.order equal to a power of the prime p.)- ~(A p-group is one in which all elements haveA group G is a divisible groupl~he author wishes to acknowledge the help of K. D. Wallace in writingthis paper.