# Determine if a group is cyclic

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1. Sep 17, 2015

### 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!

2. Sep 17, 2015

### 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?

3. Sep 18, 2015

### DeldotB

Do the powers need to divide 20?

4. Sep 18, 2015

### 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.