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Non trivial subgroups

  1. Aug 30, 2011 #1
    Prove that if a group G has no non-trivial subgroups, G is finite and o(G) is a prime number, where o(G) is the order of the group G.

    If G is infinite, you can show that there are non trivial subgroups. What remains to prove is that if o(G) is not prime, than there is at least one subgroup H, with o(H) equal to one of the prime divisor of o(G). Any idea?
  2. jcsd
  3. Aug 30, 2011 #2
    Suppose that G has no nontrivial subgroups. Take an arbitrary element x with [itex]x\neq e[/itex]. Then what can you say about <x> (the group generated by x)??
  4. Aug 30, 2011 #3
    Since G has no nontrivial subgroup and <x> is a subgroup of G, but <x> is not e, G=<x>.
  5. Aug 30, 2011 #4
    So you must only prove now that all cyclic subgroups whose order is not prime have a nontrivial subgroup...
  6. Aug 30, 2011 #5
    I prove that if G=<g> is a finite cyclic group and o(G)=m, then for every d such that d|m there is a subgroup H such that o(H)=d. It is simply this: consider the subgroup <g^(m/d)>. Since o(G)=o(g)=m, we have that o(H)=o(g^(m/d))=m/(m/d)=d. So, H is a non trivial subgroup. So, if the only possible subgroups are the trivial ones, m must be prime.
  7. Aug 30, 2011 #6
    Seems ok!! :smile:
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