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Let $G$ pe a $p$ group. I have to show that

  • the number of nonnormal subgroups is divisible by $p$
  • the number of subgroups differs from the number of normal subgroups by a power of $p$.

Are there any theorem that can help me to prove this ? We have discussed the Sylow theorems but I don't know how to apply them - if those are the theorems I need. (Does this theorem also hold for infinite groups $G$ ?)

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Have you seen the class equation? – Moderat Jan 15 '13 at 18:40
up vote 3 down vote accepted

Hint: $G$ operates on the set of subgroups by conjugation. What are the possible lengths of orbits? What does it mean if the orbit length is $1$?

Remark: We don't use finiteness of $G$ here, but the subgroup counts involved should be finite for the statement to make sense.

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If the orbit has length $1$ that means the subgroup is normal (the other direction also holds). But I don't know how to get the length of orbits for nonnormal subgroups. Could you help me there ? – user47574 Jan 15 '13 at 19:13
Orbit lengths divide the group order – Hagen von Eitzen Jan 15 '13 at 19:47

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