+20 votes
2.5k views
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
recategorized | 2.5k views
+4

This might help ...

## 1 Answer

+57 votes
Best answer

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 $5.$

answered by Active (3.7k points)
edited by
0
WHT IF A GROUP  IS AN INFINITE GROUP??
+5
concept of order exists fpr finite groups only
Answer:

+35 votes
4 answers
1
+40 votes
5 answers
2
+20 votes
5 answers
3
+29 votes
4 answers
4
+15 votes
4 answers
5
+30 votes
4 answers
6
+25 votes
5 answers
7