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A certain group G has order 100. Find a number 1 ≤ n ≤ 100 that...

A certain group G has order 100. Find a number 1 ≤ n ≤ 100 that CANNOT be the order of a subgroup H ⊆ G and briefly explain why it cannot.

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Answer #1

By Theorem of Lagrange, if H is the subgroup of a finite group G then |H| divides |G|.

Hence if d doesn't divides n then G can not have a subgroup oforder d

Here |G|=100, then posssibilities that can not be the order of a subgroup are: 3,6,7,8,9,11,12,13,14,15,16,17,18,19,21,22,23,24,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99

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