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Let $G$ be a group of $35$ elements. Then the largest possible size of a subgroup of $G$ other than $G$ itself is _______.
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Lagrange's Theorem :

Order of subgroup must be factor of Order of group.

$G$ is a group with $35$ elements. So order of $G = 35.$

Factors of $35 : 1,5,7,35$

Proper subgroup: order of the subgroup is less than the order of group.

Order of the largest possible proper subgroup $= 7.$

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Answer : 7

According to Lagrange's theorem, for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G

Note : Order = numberof elements in group

So subgroup size must be from this : 1, 5, 7, 35 (since all are divides the order of group)

Since question asking largest subgroup size other than 2 , So 7 must be answer.

Answer:

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