December 9th, 2013, 08:58 AM  #1 
Senior Member Joined: Feb 2013 Posts: 153 Thanks: 0  Sylow subgroup
Hi. I dont know how to solve following problem: Let G be the group of order p*q^2, where p and q are different prime numbers. Prove that at least one Sylow subgroup of group G is normal. Thanks. 
December 9th, 2013, 09:32 AM  #2 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Sylow subgroup
I'd approach it by noting that the number of qSylow subgroups of G must be 1, p or p^2 and analyzing each case. I am not pretty good at Sylow groups, so I'll leave it to you (or anyone else that might want to show an approach. Note that the above is not a hint as I haven't calculated up anything yet. In fact, the approach might not even work).

December 31st, 2013, 06:06 AM  #3 
Senior Member Joined: Mar 2012 Posts: 294 Thanks: 88  Re: Sylow subgroup
I will assume that you mean G = pq^2, and not (pq)^2. We will consider 3 cases. We will use P to denote a Sylow psubgroup, and Q to denote a Sylow qsubgroup. It is clear that P has 1,q or q^2 conjugates, and Q has either 1 or p conjuagtes. Case 1: p not congruent to 1 (mod q). In this case we have [G:N(Q)] = 1 (mod q), so [GN(Q)] cannot be p. Thus Q is normal in G. Case 2: p = 1 (mod q), q^2 not congruent to 1 (mod p). This implies q is not congruent to 1 (mod p), for if so: q = kp + 1, so q^2 = (kp + 1)^2 = k^2p^2 + 2kp + 1, a contradiction. Since we have [G:N(P)] = 1 (mod p), this means that [G:N(P)] is not q or q^2, hence P is normal in G. Case 3: p = 1 (mod q), q^2 = 1 (mod p). Here, q^2 = kp + 1 = k(mq + 1) + 1 = (km)q + 2, so 2 = q(q  km). This implies q2, so q = 2. Hence p = 3, so we have a group of order 12. If P (of order 3) is not normal in G, we must have 4 Sylow 3subgroups, since 2 is not congruent to 1 (mod 3). This gives us 8 elements of order 3, leaving just 4 elements left over, which must then be the sole Sylow 2subgroup of order 4. 

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