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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 __________.
asked in Set Theory & Algebra by Veteran (101k points)
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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 $5.$

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WHT IF A GROUP  IS AN INFINITE GROUP??
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concept of order exists fpr finite groups only
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