
Abstract Algebra Abstract Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
October 12th, 2017, 10:59 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 12:36 AM. 
October 13th, 2017, 12:57 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,640 Thanks: 2082 
If the problem is referring to the subgroups of some group $G$, what are you told about $G$?

October 14th, 2017, 08:55 AM  #3 
Member Joined: Jan 2016 From: Athens, OH Posts: 93 Thanks: 48 
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, 09:29 AM  #4 
Senior Member Joined: Sep 2016 From: USA Posts: 609 Thanks: 378 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 03:27 PM. 
October 15th, 2017, 02:14 PM  #5 
Member Joined: Jan 2016 From: Athens, OH Posts: 93 Thanks: 48 
SDK, I previously confessed that I can't solve problem 2. So I would be quite interested in seeing your solution. Please post it. 

Tags 
number, subgroup, subgroups 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Number of subgroups of infinite abelian group  thippli  Abstract Algebra  2  May 27th, 2013 08:26 AM 
Subgroups  gaussrelatz  Algebra  1  October 10th, 2012 11:30 PM 
Number of subgroups of Sn  honzik  Abstract Algebra  0  February 13th, 2011 06:45 AM 
Subgroups  DanielThrice  Abstract Algebra  1  November 25th, 2010 02:28 PM 
no subgroups  bjh5138  Abstract Algebra  2  August 14th, 2008 12:03 PM 