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Unformatted text preview: D for j i , we have D r i . X j x j / D X j r i x j D r i x i : Because r i is relatively prime to the order of each nonzero element of G i , it follows that x i D . Thus by Proposition 3.5.1 , G D G 1 G s . n Let G be a nite abelian group. For each prime number p dene Gp D f g 2 G W o.g/ is a power of p g : It is straightforward to check that Gp is a subgroup of G . Since the order of any group element must divide the order of the group, we have Gp D f g if p does not divide the order of G . Let p be a prime integer. A group (not necessarily nite or abelian) is called a p-group if every element has nite order p k for some k . So Gp is a p subgroup of G ....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08