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On orders of subgroups in Abelian groups: an elementary solution of an exercise of Herstein. (English) Zbl 1229.20056

Summary: I. N. Herstein’s ‘Topics in Algebra’ [New York: Blaisdell (1964; Zbl 0122.01301)] contains an exercise for which Herstein acknowledges he doesn’t know of a solution using only material developed to that point in the text. The problem is to show that, if an Abelian group \(G\) contains subgroups of orders \(m\) and \(n\), then it contains a subgroup whose order is the least common multiple of \(m\) and \(n\). This appears as Exercise 26 in Section 2.5 of the second edition of ‘Topics in Algebra’ (it is Exercise 11 in the first edition). We present a solution using material from Section 2.5 and earlier.

MSC:

20K27 Subgroups of abelian groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
97H40 Groups, rings, fields (educational aspects)

Citations:

Zbl 0122.01301
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