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October 12th, 2017, 11:59 PM  #1 
Senior Member Joined: Jan 2015 From: usa Posts: 104 Thanks: 1  Number of subgroups
Please help me to prove the following problem: >1) Every subgroup of order $p^2$ contains $1\;(\text{mod }p)$ different subgroups of order $p$. >2) Every subgroup of order $p$ is contained in $1\;(\text{mod }p)$ different subgroups of order $p^2\!$. Thanks. Last edited by skipjack; October 13th, 2017 at 01:36 AM. 
October 13th, 2017, 01:57 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,264 Thanks: 1959 
If the problem is referring to the subgroups of some group $G$, what are you told about $G$?

October 14th, 2017, 09:55 AM  #3 
Member Joined: Jan 2016 From: Athens, OH Posts: 92 Thanks: 47 
Your last couple of posts are related to a theorem of Helmut Wielandt. If G is a finite group of order $p^kn$ where p is a prime, then the number of subgroups of G of order $p^k$ is congruent to 1 mod p. Here, it is not assumed that p is prime to n. A proof can be found here https://math.stackexchange.com/quest...sylowstheorem Now as to your specific questions (I assume p is prime and $p^2$ divides the order of G.) 1. Let H be a subgroup of order $p^2$. Either H is cyclic or H is the direct product of 2 subgroups of order p. In the first case, H has exactly one subgroup of order p. In the second case, H has $p^21$ elements of order p and hence ${p^21\over p1}=p+1$ subgroups of order p. 2. I can't prove this without Wielandt's theorem. However, it is true and here's a proof. Let H be a subgroup of order p and $S=\{K:H\subset K\text{ and K is a subgroup of order }p^2\}$. Let L be the subgroup generated by all the elements of S (S is not empty). Since each $K\in S$ is abelian, H is a normal subgroup of L. By Wielandt, the number of subgroups of L/H of order p is 1 mod p. That is, the cardinality of S is 1 mod p. 
October 14th, 2017, 10:29 AM  #4 
Senior Member Joined: Sep 2016 From: USA Posts: 555 Thanks: 319 Math Focus: Dynamical systems, analytic function theory, numerics 
Here are some hints: Start by proving that any group of order $p^2$ is abelian. To prove this consider such a group mod its center. What can you say if this quotient group is cyclic? Now apply Sylow's theorem to any group of order $p^2$. For 2, I would suggest looking at equivalence classes of an element of order $p$ under conjugation. Apply the orbit stabilizer theorem to count the ways an element of order $p$ can belong to a subgroup of order $p^2$. Last edited by skipjack; October 14th, 2017 at 04:27 PM. 
October 15th, 2017, 03:14 PM  #5 
Member Joined: Jan 2016 From: Athens, OH Posts: 92 Thanks: 47 
SDK, I previously confessed that I can't solve problem 2. So I would be quite interested in seeing your solution. Please post it. 

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