Let G be a finite group on 84 elements. The size of a largest possible proper subgroup of G is _____

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Question

Let G be a finite group on 84 elements. The size of a largest possible proper subgroup of G is _______.

Accepted Answer

Correct Answer: 42Step-by-step Solution:By Lagrange's Theorem, the order (size) of any subgroup of a finite group G must divide the order of G. Here |G| = 84.Factorize: 84 = 2² × 3 × 7. The divisors of 84 are: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.A proper subgroup must have order strictly less than 84. The largest proper divisor of 84 is 42 = 84/2.A subgroup of index 2 (i.e., of order |G|/2) always exists and is always normal. Therefore, G has a subgroup of order 42, which is the largest possible proper subgroup.Key Concept: Lagrange's Theorem states that the order of every subgroup of a finite group G divides |G|. The largest proper subgroup has order equal to the largest proper divisor of |G|, which is |G| divided by its smallest prime factor. Since the smallest prime factor of 84 is 2, the largest proper subgroup has order 84/2 = 42. A subgroup of index 2 is guaranteed to exist and is always normal.

Let G be a finite group on 84 elements. The size of a largest possible proper subgroup of G is _______.

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Let G be a finite group on 84 elements. The size of a largest possible proper subgroup of G is _______.

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