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Subgroup of given order of an Abelian group

  1. May 17, 2010 #1
    We know that if there exists an element of a given order in a group, there also exists a cyclic subgroup of that order. What about converse?
    Suppose there is a subgroup of an Abelian group of order 'm'. Does that imply there also exists an element of order 'm' in the Group. It does not hold in general for Non-Abelian groups. But what about Abelian groups?
  2. jcsd
  3. May 17, 2010 #2


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    Suppose that were true. what if m = |G|?
  4. May 17, 2010 #3


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    Take a look at an elementary Abelian 2 -group of order greater than 2 e.g. Z/2 x Z/2 x Z/2.
  5. May 17, 2010 #4
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