Yes, there is a group with five elements that is not isomorphic to $\mathbb{Z}_5$ (the cyclic group of integers modulo 5).

The key point is that groups of order 5 are either cyclic or non-cyclic. However, a group of order 5 can only be cyclic by Lagrange's Theorem, because the order of every element must divide the order of the group. In particular, a finite group of prime order must be cyclic because every non-identity element must generate the entire group.

So, for a group to have 5 elements and be non-isomorphic to $\mathbb{Z}_5$, it would need to fail to be cyclic, but there is no such group. All groups of order 5 are cyclic, and all cyclic groups of order 5 are isomorphic to $\mathbb{Z}_5$.

Therefore, every group of order 5 is isomorphic to $\mathbb{Z}_5$. No group of order 5 is non-isomorphic to it.

Related questions:

1. What makes a group cyclic or non-cyclic?
2. What is Lagrange's Theorem and how does it help classify groups?
3. Can there be a non-abelian group of order 5?
4. What is the classification of groups of prime order?
5. How do we prove that $\mathbb{Z}_5$ is cyclic?

Tip:

When working with groups, always check the order of the group and the possible orders of its elements. If the group order is prime, it's guaranteed to be cyclic.