I recently had a homework problem in my Analysis course that went as follows:
Show for any $n\in\mathbb{Z}$, then either $n$ is even, i.e. $n= 2k$ for some $k\in\mathbb{Z}$, or $n$ is odd, i.e. $n = 2k + 1 $ for some $k \in \mathbb{Z}$.
This seems easy enough however there was a problem with this. Our definition of $\mathbb{Z}$ was only the axioms of an abelian group. This introduced the problem of $1$ not being defined, so I got around this by modifying the question to be
Show there exists $a$ such that for any $n\in G$, then either $n$ is even, i.e. $n= 2k$ for some $k\in G$, or $n$ is odd, i.e. $n = 2k + a $ for some $k \in G$.
This new statement is provable on some groups, but not others. I proved it was the case for cyclic groups and handed in my homework.
However this got me thinking since not all groups have this property what groups do? What happens when we consider groups in general and not just abelian groups? What statements can we make about groups that do and do not have this property? I've begun to call these Even or Odd groups, so I will use this throughout this question. If you think you have a better name I'm open to that sort of suggestion.
I've proven a couple of facts myself about these groups.
The first thing I proved was that cyclic groups are all Even Odd groups. Here is the proof I provided of this:
Let $n$ be an element of a cyclic group, let $0$ be the group's identity and $a$ its generator.
If $n$ is even $n+a$ must be odd, because $n$ is representable as $k+k$ by definition thus $n+a$ is representable as $k+k+a$ the definition of odd.
If $n$ is odd $n+a$ must be even, because $n$ is representable as $k+k+a$ by definition thus $n+1$ is representable as $k+k+a+a$. Since all cyclic groups are abelian this is the same as $(k+a)+(k+a)$ showing that $n+a$ must be even.
It is the case that $0+0 = 0$ because $0$ is the identity. This makes $0$ even.
By induction we know that all members of our group are either even or odd.
After that I proved that self-inverting groups of size greater than 2 cannot be Even or Odd groups.
We know that $\forall k \in G : k+k = 0$. This means that $0$ is the only even number and in addition no matter what we define $1$ as there can be at most a single odd number.
If the group has a size greater than 2, via the pigeon hole principle there must be a member that is neither even nor odd.
I then realized we could make the more general statement, that if less than half of the group is even the group cannot be an Even Odd group. This proof follows pretty neatly from the last so I will forgo including it.
It seemed natural that I would try to prove the inverse of this, that is are all groups that have at least half their elements even, Even or Odd groups? However I was neither able to find a counter example or to prove this.
So what statements can we make about even or odd groups? Are all groups that have at least half their elements even, Even or Odd groups?
