Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than G itself is ________ .
Note – This question was Numerical Type.
(A) 1
(B) 5
(C) 7
(D) 35
Answer: (C)
Explanation: According to the Lagrange’s Theorem:
“If H is a subgroup of finite group G then the order of subgroup H divides the order of group G.”
Order of subgroup must be factor of Order of group.
Since order of a group is number of elements present in that group, therefore order of given group will be 35.
Factor of 35 are: 1, 5, 7, 35.
So, largest possible size of a subgroup of G other than G itself is 7.
Size of group = Ο(G) = 35
Let H be subgroup of G
∴ Ο(H)⏐Ο(G)
Possible orders of H are 1, 5, 7, 35
Size of largest possible proper subgroup = 7.
Option (C) is correct.
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