GATE CSE 2020 | Question: 18

Question

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

Comments

What book/resource should we use to study this topic ? (Group theory)

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rosen hard copy

Answer 1

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 $= 7.$

Comments

how Smallest possible Proper subgroup is 3 it should be 1 ?

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1)Smallest possible subgroup order: 1

2)Smallest possible Proper subgroup order : 1 or 3 depends (check below Wikipedia link)

3)Largest possible subgroup order : 35

4)Largest possible proper subgroup order : 7

 

https://en.wikipedia.org/wiki/Subgroup

The trivial subgroup of any group is the subgroup {e} consisting of just the identity element.

A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G). This is usually represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (that is, H ≠ {e}).

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@rajankakaniya but how is 35 divisible by 3, I am not able to understand. The smallest size possible is 5 (for proper subgroup) while 1 (for subgroup). Please help.

Answer 2

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 3

size of a subgroup has to divide the size of the group evenly.

factors of 35 are : 1,5,7,35

Given subgroup other than G, Therefore largest subgroup possible is of size '7'

Answer 4

7

Total subgroup of G are 1, 5,7,35

Largest possible size of subgroup except G is size of 7 element 

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