GATE CSE 2014 | Set 3 | Question: 3

Question

Let $G$ be a group with $15$ elements. Let $L$ be a subgroup of $G$. It is known that $L \neq\ G$ and that the size of $L$ is at least $4$. The size of $L$ is __________.

Answer 1

Lagrange's theorem: For any finite group $G,$ the order (number of elements) of every subgroup $L$ of $G$ divides the order of $G.$
$G$ has $15$ elements.
Factors of $15$ are $1,3,5,$ and $15.$
Since, the given size of $L$ is at least $4$ $(1$ and $3$ eliminated$)$ and not equal to $G(15$ eliminated$),$ the only size left is  $5.$

Size of $L$ is $5.$

Comments

WHT IF A GROUP  IS AN INFINITE GROUP??

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concept of order exists fpr finite groups only

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Lagrange’s Theorem is for finite groups.

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got it

Answer 2

Lagrange’s theorem specifies: Order of subgroup divides the order of a group order of a group= number of elements in a group = 15

L is a subgroup

L ⊆ G and L ≠ G 

Divisor of 15 = 1, 3, 5, 15

L is at least 4, that is L ≥ 4

Therefore, The size of L is 5

Answer 3

 

Lagrange's  one way theoram :(in case of group) if h is subgroup of G then order of h will divide order of G.

while its converse is not true in case of group that if let say some d divides G then subset of size d will be subgroup .

but this converse is true for the finite abelian group 

credit @GO Classes