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The problem I am struggling with is
"Find an example of a finite group G of order n where n is odd and each element of G is a square".
In my understanding this means
"Find a finite group with odd number of elements where each element can be expressed in ( )^2 form".
Am I understanding the question incorrectly?
I can't seem to find any finite group that fits this criteria.
Could anybody help me?
Let $ G $ be a finite group of odd order. Let $ \phi: G \to G $ via $ x \mapsto x^2 $. Then we claim $ \phi $ is an injection.
Let $ |G| = n $. Suppose $ x^2 = y^2 $. $ x = x^{n + 1} $. Then $ n + 1 $ is even, so let $ n + 1 = 2k $. Similarly, $ y = y^{2k} $. Then $ x = x^{2k} = y^{2k} = y $. Thus $ \phi $ is an injection. As $ G $ is finite, it is therefore a surjection, so every element of $ G $ is a square. Thus, all groups of odd order satisfy this.
If $|G| = n$ then for any $g\in G$, $g^n = e$. So $g^{n+ 1} = g$ and $g= (g^{\frac {n+1}2})^2$ is a square.
So all groups with odd order satisfy this condition.
So you couldn't find any groups that failed the condition. You just didn't realize how they satisfied it.
....
(One thing to note. If the operation is "$*$" then a square is a number $m = x*x$. So if the operation is addition a "square" is a number $m = x + x$. So for, say, $\mathbb Z_5, +$ $0 = 0+0; 1 = 3+3; 2=1+1; 3=4+4; 4=2+2$ are all squares.
