Is it true that a group is divisible if and only if it has no maximal subgroup ?

I need to find what are the at the group $\mathbb{Q}/\mathbb{Z}$. I think that any element at this group has a finite order, but I don’t know how to prove it… I’d like to get help with the proof writing… If I’m wrong, I’d like to to know it too… BTW: $\mathbb{Z}=(\mathbb{Z},+)$ $\mathbb{Q}=(\mathbb{Q},+)$ Thank you!

The title says it all … Formally, if $SO_n(\mathbb R)=\lbrace A\in M_n({\mathbb R}) |AA^{T}=I_n, {\sf det}(A)=1 \rbrace$ and $W\in SO_n(\mathbb R)$, is it true that for every integer $p$, there is a $V\in SO_n(\mathbb R)$ satisfying $V^p=W$ ? This is obvious when $n=2$, because rotations in the plane are defined by an angle which can […]

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