Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than G itself is _______.

Question

Accepted Answer

7 By Lagrange's theorem, every subgroup's order must divide the group's order. For 35 = 5 * 7, the proper divisors are 1, 5, and 7. The largest proper subgroup thus has order 35 divided by the smallest prime factor (5), giving 7. Any group of prime power order is well-understood, and a subgroup of order 7 always exists by Cauchy's theorem.