Categoria: Abstract Algebra
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The “towers” method. Finding the number of elements with a given order in an Abelian group with cardinality a power of a prime.
Suppose we have an Abelian group G of cardinality p^n, where p is a prime number. By the classification theorem for Abelian groups, we already know that G is isomorphic to a direct product of cyclic groups (whose order will be, clearly, a power of p). So we know how to list all possible group…
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Groups of order 18
Anyone who has taken a course in Group Theory has encountered this type of exercise: listing all groups of a given order (up to isomorphisms). In this article, I want to summarize some of the most commonly used techniques for answering this question, applying them to the case of groups of order 18. Sylow’s theorem…