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Introduction A group $G$ is solvable if there exists a chain of subgroups \(\lbrace e\rbrace =G_n\lhd G_{n-1}\lhd \cdots \lhd G_1 \lhd G_0 =G\) such that each $G_{i-1}/G_i$ is cyclic. Theorem. For finite $G$, the definition of solvable is not altered when the condition that each quotient group is cyclic is replaced by being abelian. Proof On...

Introduction We will prove that the regular heptadecagon, a polygon with 17 equal sides, is constructible with ruler and compass. It is enough to prove that the number $\cos\left(\frac{2\pi}{17}\right)$ is constructible. The procedure will follow the idea given by Gauss, who summed the roots of unity to express this quantity in terms of roots—G...

Introduction The following theorem—to the best of my knowledge—was proven by Reinhold Baer in 1936 [1], and Irving Kaplansky, in 1954 [2], was the first to publish a proof using Zorn’s lemma, which is the one we will follow below. Theorem. Let $A$ be an abelian group defined additively. Suppose that $A$ is divisible, that is, for every $a\in A...