TITLE: Minimum possible order of a group with elements of order 1 to 5
QUESTION [7 upvotes]: If G is a group containing elements of order 1 to 5, then what is the minimum possible order of this group?

My answer: We know that the order of the elements of a group divides the order of group, so the minimum possible order of G would be the l.c.m of 1, 2, 3, 4 & 5 which is 120.
I just wanted to confirm if I am right.

REPLY [11 votes]: Consider $Z_{60}$, the cyclic group of order 60. If it is generated by $x$, then $x^0$ has order 1, $x^{30}$ has order 2, $x^{20}$ has order 3, $x^{15}$ has order 4, and $x^{12}$ has order 5. Note that by your argument, we know that the lcm of 1, 2, 3, 4 and 5 must divide the order of such a group, and since the lcm is 60, the minimum possible order is 60, and indeed, as we just showed, there is a group with this property of this order.