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# GATE | Sudo GATE 2020 Mock I (27 December 2019) | Question 17

• Last Updated : 26 Dec, 2019

Let G be a finite group on 84 elements, then which of the following option is True?
(A) The size of a largest possible subgroup of G is 42.
(B) The size of a smallest possible proper subgroup of G is 2.
(C) The size of a smallest possible proper subgroup of G is 1.
(D) The total number subgroups of group G are 16.

Explanation: According to Lagrange’s theorem, states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G.

Therefore, possible subgroups of group on 84 elements are : 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.

Subgroups with element 1 and 84 are trivial groups.

The size of a largest possible proper subgroup of G is 42.

Therefore, only option (B) is true.

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