TITLE: Let G be a finite group that has elements of every orders from 1 - 12. What is the smallest possible value of |G|
QUESTION [1 upvotes]: Let G be a finite group that has elements of every orders from 1 through 12. What is the smallest possible value of |G|
I need your help

REPLY [3 votes]: Since the order of every element divides the order of the group, $G$ must be at least the lowest common multiple of the numbers $1$ to $12$. This number turns out to be $m=2^3\cdot3^2\cdot5\cdot7\cdot11 = 27720$.
If you look at $\mathbb{Z}_m$, you can easily see that in fact it has elements of all orders from 1 to 12. So $27720$ is the answer.

REPLY [1 votes]: By Lagrange, it would have to have order at least the LCM of the integers from 1-12. Can you think of a group of this order, and does it have elements of all of these orders?

REPLY [0 votes]: To add to previous answers and comments, note that the cyclic group $\mathbb{Z}_m$ has an element of order $n \iff n|m$.  Can  you prove this?  Would such a group be the smallest group?
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