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Sylow's Theorems and Simple Groups

  1. Nov 17, 2012 #1
    I am wondering if some one can help it this:
    Suppose G is a group with 316 [itex]\leq[/itex]|G|[itex]\leq[/itex] 325. Given that G is simple, find the possible value(s) for |G|. Be sure to explain your reasoning for each number. You'll need Sylow's Theorems of course.

    This is what I have done:
    the prime factorization of |G|:

    316 2 2 79
    317 317
    318 2 3 53
    319 11 29
    320 2 2 2 2 2 2 5
    321 3 107
    322 2 7 23
    323 17 19
    324 2 2 3 3 3 3

    If p is the largest prime factor, and |G| = mp^k where p doesn't divide m, the the p-Sylow subgroup is normal (The number of p=Sylow subgroups, n_p, = 1).

    All the results except for 320 and 324 are straightforward:

    316 n79 = 1
    317 G is Z/317 and simple
    318 n53 = 1
    319 n29 = 1

    321 n107=1
    322 n23 = 1
    323 n19=1

    Thanks in advance
     
  2. jcsd
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