Let \(G\) be a group with 15 elements. Let \(L\) be a subgroup of \(G\). It is…
2014
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 __________.
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Correct answer: 5
Key insight: the order of any subgroup divides the order of the group (Lagrange's theorem).
The possible sizes of a subgroup of a group of order 15 are the divisors of 15: 1, 3, 5, 15.
The subgroup is not the whole group, so its size is not 15.
The size is given to be at least 4, which rules out 1 and 3.
Therefore the only possibility left is 5.
Answer: 5.
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