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You are not logged in. #1 20090517 08:56:02
Simple groups (2)This is a spinoff from a thread in this section, in which it was proved that all finite groups of order ≤ 120 are not simple, except the trivial group, the primeordered groups and a certain group of order 60. Not simple. Such a group, being of order , is Abelian and has a subgroup of order . Not simple. The Sylow 61subgroup has index 2 and is therefore normal. Not simple. The Sylow 41subgroup is unique and therefore normal. Not simple. The Sylow 31subgroup is unique and therefore normal. Not simple. These are 5groups and so have nontrivial centres. Not simple. The Sylow 7subgroup is unique and therefore normal. Simple. Order is prime. Not simple. These are 2groups and so have nontrivial centres. Not simple. The Sylow 43subgrouop is unique and therefore normal. Not simple. The Sylow 13subgrouop is unique and therefore normal. Simple. Order is prime. Not simple. See post #4. Not simple. Group of order pq, p ≠ q. See post #6. Not simple. Group of order pq, p ≠ q. See post #6. Not simple. The Sylow 5subgrouop is unique and therefore normal. Not simple. The Sylow 17subgroup is unique and therefore normal. Simple. Order is prime. Not simple. Sylow 23subgroup is unique and therefore normal. Simple. Order is prime. Not simple. Sylow 7subgroup is unique and therefore normal. Not simple. Group of order pq, p ≠ q. See post #6. Not simple. Group of order pq, p ≠ q. See post #6. Not simple. Group of order pq, p ≠ q. See post #6. Last edited by JaneFairfax (20090525 21:58:42) #2 20090517 11:45:09
Re: Simple groups (2)I would suggest to try to come up with arguments that handle a large number of orders, rather than go one by one. For example, any group of order pq will not be simple. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #3 20090517 22:54:29
Re: Simple groups (2)Yes, I’m already familiar with a lot of results that apply to various classes of orders – thanks mainly to you and Humphreys. Most orders I encounter when going through the orders one by one usually fit one of these familiar results, so I simply quote the relevant result and say, “That’s that order done.” #4 20090518 22:48:27
Re: Simple groups (2)Let . There are 1 or 12 Sylow 11subgroups. If 12, then the union of these subgroups has elements and so there can’t be 22 Sylow 3subgroups in this case. So there are 1 or 4 Sylow 3subgroups. If 4, then the union of all the Sylow 3 and 11subgroups has elements. The remaining 3 elements must therefore be the nonidentity elements in the unique Sylow 2subgroup. Hence a group of order 132 is not simple. #5 20090519 07:28:21
Re: Simple groups (2)Jane, let's just continually modify your opening post, keep them all in one place. If we have a general argument that works for multiple orders, or if an argument for a certain order is long, make a post below it and then just reference that post number. Also, please write the prime factorization for each order in the list! "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #6 20090519 07:38:34
Re: Simple groups (2)The number of Sylow p groups must divide q and be congruent to 1 (mod p). The only such number is 1, and thus the Sylow psubgroup is normal. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #7 20090519 18:55:50
Re: Simple groups (2)
That’s fine with me. #8 20090525 22:05:14
Re: Simple groups (2)I’d just like to point out that there is a simple group of order 168, namely the projective special linear group of degree 2 over a field with 7 elements. #9 20090526 06:57:54
Re: Simple groups (2)"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." 