I'm just trying to grasp ordering of groups and subgroups a little better,
I get the basics of finding the order of elements knowing the group but I have a few small questions,
If you let G be a group of order 100, what are the possibilities for the order of g12?
Would they just be divisors of 100? 2, 5 , 10, 20, 25 and 50?
Or would the power of g be restricted to below 100, giving orders of only 2 and 5?
Another question I have regards the order of an intersection of subgroups,
Say if you have a group with two subgroups A and B, where the order of A is 120 and the order of B is say 105,
The lowest common multiple of these two numbers is 840,
But if you take the intersection between the two subgroups A and B, what would be the possible orders of the intersection?
Would it just be multiples and divisors of 840 which are less then the order of the group?
Thanks in advance