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 ²±±±² G s . n Let G be a ﬁnite abelian group. For each prime number p deﬁne GŒpŁ D f g 2 G W o.g/ is a power of p g : It is straightforward to check that GŒpŁ is a subgroup of G . Since the order of any group element must divide the order of the group, we have GŒpŁ D f g if p does not divide the order of G . Let p be a prime integer. A group (not necessarily ﬁnite or abelian) is called a p-group if every element has ﬁnite order p k for some k µ . So GŒpŁ is a p –subgroup of G ....
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- Fall '08
- Algebra, Prime number, Abelian group, Cyclic group, finite abelian group