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

@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.

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wait how 3 will come 3 dont divide 35 right?

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I think it's a typo 3 does not divides 35

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

7

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

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

Same type questions

https://gateoverflow.in/204093/gate2018-19

https://gateoverflow.in/2037/gate2014-3-3

https://gateoverflow.in/226421/group-theory

Answer 4

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 5

1. If S is subgroup of G then |S| divides |G| 

2. So the largest factor (which is Not equal to 35) is 7.  

Answer is 7