Let p be a prime number. Then it is known that every group of order p2 is abelian. The following is a proof which does not resort to contradiction.
Tag: Abstract algebra
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The classical groups are one of the major sources of interest in algebra and topology. When one replaces the field of real numbers by the ring of integers or the ring of integers mod N, they are still of interest in algebra.
Here we consider the relationship between SL (2, Z) and SL (2, Z / NZ).
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The correspondence theorem is one of the isomorphism theorems in algebra valid for both pairs (a group and its normal subgroup) and (a commutative ring and its ideal). Unfortunately, not many textbooks on algebra (or group theory or ring theory, for that matter) offer a detailed proof. Thus, this article may be of some use to students of mathematics.
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This is the result of the site owner’s struggle to understand algebra (i.e., theory of groups, rings, fields, etc.) with the starting point based on Modern Algebra by professor Akira Hattori.
There is no claim for originality; in fact, it is a hodge-podge of citations from literally hundreds of textbooks and articles.
Caveat: The chapter 12 Algebraic K-theory is only sketchy.
MMM 