Can a Group of Order 20 with Elements of Order 4 Be Cyclic?

[DeldotB]

Hello all!

If I have a group of order 20 that has three elements of order 4, can this group be cyclic? What if it has two elements? I am new to abstract algebra, so please keep that in mind!

Thanks!

 

[andrewkirk]

If it's cyclic then it has a generator element g such that $g^{20}=1$ and $1,g,g^2,...,g^{19}$ are all different.

Let the three elements of order 4 be a, b and c.

What can we deduce about what powers of g each of those elements could be?

 

[DeldotB]

Do the powers need to divide 20?

 

[andrewkirk]

That's a sufficient, but not a necessary condition.

Think about the* cyclic group of order 20: {1,$g,g^2,...,g^{19}$}. Express the fourth power of each of its elements as $g^m$ where $m<20$.

*Note the use of 'the' rather than 'a'. All cyclic groups of order 'n' are isomorphic.