Masters Examination in Mathematics

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Solution. Let np denote the number of p-Sylow subgroups. Since n11 is 1 mod 11 and divides 12, we know n11 is either 1 or 12. If it is 1, then it is normal and we are done. So instead, assume n11 is 12. Then there must be 120 elements of order 11 in G, leaving only 12 more elements available. Since n3 is 1 mod 3 and divides 44, we know that n3 is 1, 4 or 22. If it is 1, then again it is normal and we would be done. If it is 22, that would require 44 elements of order 3 which is not possible. If n3 is 4 that only requires 8 elements of order 3. This leaves 4 elements available which must form the unique Sylow 2-subgroup, which is then normal.

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تاریخ انتشار 2011