All groups of order 99 are abelian.

[JFo]

Prove all groups of order 99 are abelian:

I'm stuck right now on this proof, here's what I have so far.

proof:
Let G be a group such that |G| = 99, and let Z(G) be the center of G.
Z(G) is a normal subgroup of G and |Z(G)| must be 1,3,9,11,33, or 99.

Throughout I will make repeated use of the theorem which states if the factor group G/Z(G) is cyclic, then G is abelian.

Case 1:
Assume |Z(G)| = 99, then Z(G) = G, and G is abelian.

Case 2:
Assume |Z(G)| = 33, then |G/Z(G)| = 3, a prime, so G/Z(G) is cyclic, and thus G is abelian.

Case 3:
Assume |Z(G)| = 9, then |G/Z(G)| = 11, a prime, so G/Z(G) is cyclic and G is abelian.

Case 4:
Assume |Z(G)| = 3, then |G/Z(G)| = 33 which factors into (3)(11). There is a theorem which states that if a group is order of a product of two distinct primes p,q with p<q, then G is cyclic if q is not congruent to 1 modulo p. Since 11 is not congruent to 1 mod 3 G/Z(G) is cyclic, and so G is abelian.

OK, here's where I get stuck!

Case 5:
Assume |Z(G)| = 11, then |G/Z(G)| = 9 = 3^2. Since G/Z(G) is order of a prime squared, G/Z(G) is abelian. Thus by the theorem of finitely generated abelian groups, G/Z(G) is either isomorphic to Z_9 or Z_3 x Z_3. If its isomorphic to Z_9, then G/Z(G) is cyclic and were done. But if its isomorphic to Z_3 x Z_3 then I don't know how to proceed.

Case 6:
Assume |Z(G)| = 1, the |G/Z(G)| = 99. I'm not sure how to proceed from here.

Any suggestions?

 

[matt grime]

How many Sylow-3 subgroups does a group of order 99 have? How many Sylow-11 subgroups?

 

[JFo]

There is only 1 Sylow 3-subgroup and 1 Sylow 11-subgroup in a group of order 99. Denote these as S_3 and S_{11}.
|S_3| = 9 and |S_{11}| = 11.
S_{11} is cyclic and and every nonzero element of S_{11} is of order 11. This implies that S_3 \cap S_{11} = \{e\}.
Therfore |S_3S_{11}| = 99 so S_3S_{11} = G.
Also G \simeq S_3 \times S_{11}.
S_3 is order of a prime squared and thus is abelian, and S_{11} is abelian because it is cyclic. Therefore S_3 \times S_{11} is abelian and hence G is abelian. QED

Is this right?

Wow, thanks much for your help Matt!

 

[matt grime]

When ever you're given some group of order a small product of 2 primes (or possibly three primes) it is almost always going to be the case that looking at Sylow subgroups will help. Eg show that every group of order pqr where p<q<r are primes is solvable.

 

[JFo]

Thanks, that's useful advice.

Now that I think of it though, the proof above doesn't rely on the fact that S_3 and S_11 are Sylow p-subgroups. I could have just stated that there exist subgroups of orders 9 and 11 in G (since 9 and 11 are powers of a prime dividing |G|), and the same results would follow. Is there another proof you had in mind when you gave that hint?

 

[matt grime]

No, you can't just conclude that. You need to know that both subgroups are normal, and you do that because you can count the number of conjugates of them using Sylow's theorems.Consider a group of order 6. It has a subgroup of order 2 and a subgroup of order 3 (whcih is normal) but it is not necessarily isomorphic to C_3xC_2.

And how are you going to state that there is a subgroup of order 9 in G if you aren't going to use the fact that it is a sylow subgroup?

 

[JFo]

Oh, your right. I got two theorems mixed up. Thanks again for your help

 

[StatusX]

Not really relevant, but I always hated the wording of the theorem about G/Z(G) being cyclic implying G is abelian. If G is abelian, Z(G)=G and G/Z(G) = 1. So a much better way to state this theorem is that G/Z(G) may not be a cyclic group of order greater than one.