Question

Let K be a group with 8 elements. Let H be a subgroup of K and H<K. It is known that the size of H is at least 3. The size of H is __________

(a) 8

(b) 2

(c) 3

(d) 4

This question was addressed to me in an international level competition.

My question is from Groups topic in portion Groups of Discrete Mathematics

Answer 1

Correct answer is (d) 4

The best I can explain: For any finite group G, the order (number of elements) of every subgroup L of G divides the order of G. G has 8 elements. Factors of 8 are 1, 2, 4 and 8. Since given the size of L is at least 3(1 and 2 eliminated) and not equal to G(8 eliminated), the only size left is 4. Size of L is 4.