# What kinds of groups are there where every (nontrivial) element has prime order?

What kinds of finite groups satisfy the condition that the orders of all their (nontrivial) elements are prime? Is there a way to classify such groups?

Such groups definitely exist ($\mathbb{Z}^{n}_p$ (all its elements have prime order $p$ and this class of groups describes all abelian groups satisfying that condition (due to structure theorem for finite abelian groups)) and they can even be non-abelian (Discrete Heisenberg Group $H_{3}(\mathbb{Z}_p)$ (all all its elements have prime order $p$)). However, I do not know any other examples.

Moreover, I failed even to make out, if the orders of each element of the group should be equal, or there is a group, satisfying both that condition and the condition, that there exist two its elements with distinct prime orders. All I know, is that if such a group exists, it has to be non-abelian (as any finite abelian group with two elements of distinct prime orders $p$ and $q$ has an element of order $pq$ and $pq$ is not prime)

Any help will be appreciated.

## 1 Answer

You can take a look at the paper Classification of Finite Groups with all Elements of Prime Order by Marian Deaconescu, which seems to definitively solve your problem. The answer is apparently that for such a group $G$, we must have one of the following:

1. $G$ is a $p$-group of exponent $p$

2. $G$ is a non-nilpotent group of order $p^aq$

3. $G\cong A_5$