I recently tried to find an answer to the following question:

"Consider a group where every element is self inverse. What are the possibilities for the size of the group?"

For various reasons, I believe the answer is that any group where every element is self inverse must have order $2^n$ for some $n$.

We can construct examples of such groups by taking the product of cyclic groups of order $2$. Such examples have size $2^n$. It turns out that, up to isomorphism, these are in fact the only groups satisfying the condition (I think).

This lead me to another question; what happens when we take the product of cyclic groups which all have order 3? Or order 4? Do we get other interesting properties?

This is easy enough to play around with, but I'd like to know whether this type of group (a group which is the direct product of n cyclic groups, each of order m), has been studied? Is there a name for this type of group?

If the above questions are not immediately known to anyone, ideas/information would be extremely welcome on whether studying such groups should be in any way interesting - or alternatively completely fruitless and boring.

  • $\begingroup$ When $n$ is prime, these groups are called elementary abelian. They are very well-behaved groups. $\endgroup$ – pjs36 Dec 21 '16 at 17:43

$\newcommand{\GL}{\mathrm{GL}}$You are speaking of the so called homocyclic groups.

A finite homocyclic group $G$ which is the product of $k$ cyclic groups of order $n$ is the free group in $k$ generators in the variety of abelian groups of exponent $n$.

Among their properties, I would single out one related to the freeness. If $n = p^e$ is a power of the prime $e$, then the automorphism group $G$ induces the whole general linear group $\GL(k, p)$ on the quotient $G/G^ p$.


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