Discuss the nature and number of 3-sylow group and 5-sylow group
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hey mate here ur answer
There are several things that confuse me about this proof, so I was wondering if anybody could clarify them for me.
Lemma Let G be a group of order 3030. Then the 55-Sylow subgroup of G is normal.
Proof We argue by contradiction. Let P5P5 be a 55-Sylow subgroup of G. Then the number of conjugates of P5P5 is congruent to 1 mod 5 and divides 6. Thus, there must be six conjugates of P5P5. Since the number of conjugates is the index of the normalizer, we see that NG(P5)NG(P5) = P5P5.
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There are several things that confuse me about this proof, so I was wondering if anybody could clarify them for me.
Lemma Let G be a group of order 3030. Then the 55-Sylow subgroup of G is normal.
Proof We argue by contradiction. Let P5P5 be a 55-Sylow subgroup of G. Then the number of conjugates of P5P5 is congruent to 1 mod 5 and divides 6. Thus, there must be six conjugates of P5P5. Since the number of conjugates is the index of the normalizer, we see that NG(P5)NG(P5) = P5P5.
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