abstract algebra

tyrannosaurus

1. The problem statement, all variables and given/known data
Suppose that G is a group with exactly eight elements of order 10. How many cyclic subgroups of order 10 does G have?

2. Relevant equations



3. The attempt at a solution
I really don't have a clue how to solve this, any help would be greatly appreciated.

Dick

If you have one cyclic subgroup of order 10, how many elements of order 10 does that force G to have?

tyrannosaurus

you would need four elements of order 10 (by the Euler phi function). does that mean there would be two cyclicsubgroups of order2?

Dick

You tell me. You definitely need more than one. Can two cyclic subgroups of order 10 share any elements of order 10?

tyrannosaurus

there would only be 2 cyclic subgroups of order 10 because non of the subgroups can share an order 10 element because if they did share an element in common, that element would generate both groups, so the two groups would be the same. So this means that no two cyclic subgroups of order 10 can share an element of order 10.
Thanks a lot for the help, this made sooooooo much more sense now.