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\title{The Fundamental Theorem of Finite Abelian Groups}

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\bt {\bf The Fundamental Theorem of Finite Abelian Groups:} Every finite abelian group is the direct sum of cyclic groups of prime power order.
\et

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\visible<2->{\item<beamer:check@8-> In any group, the set of elements whose order is a power of a given prime $p$ is a subgroup of that group, denoted $G(p)$.}
\visible<3->{\item<beamer:check@9-> In abelian groups, each element of finite order is a sum of elements of prime power order.}
\visible<4->{\item<beamer:check@10-> In \textit{finite} abelian groups, this sum is unique.}
\visible<5->{\item<beamer:check@11-> If $|G|=p_1^{e_1}\cdots p_k^{e_k},$ then combining 1, 2, and 3 gives $G=G(p_1)\oplus \cdots \oplus G(p_k).$}
\visible<6->{\item<beamer:check@12-> If $p$ is prime, then $G(p)$ can be written as $<a>\oplus K$ for some $a\in G(p)$ and $K\le G(p).$}
\visible<7->{\item<beamer:check@13-> Induct!}
\ee



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