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\title{Uniqueness for FTFAG}

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\begin{frame}\frametitle{Strategy (Uniqueness for FTFAG)}

\bt Let $G$ and $H$ be finite abelian groups. Then $G$ is isomorphic to $H$ if and only if $G$ and $H$ have the same elementary divisors.
\et

\be
\visible<2->{\item<beamer:check@8-> Show $\varphi(G(p))\cong H(p)$ for each prime $p$ so that it suffices to prove the theorem for $p$-groups.}
\visible<3->{\item<beamer:check@9-> Induct on the order of the groups.}
\visible<4->{\item<beamer:check@10-> Consider $pG$ so as to take advantage of the induction hypothesis.}
\visible<5->{\item<beamer:check@11-> Identify the direct summands (and elementary divisors) for $pG$ and $pH$ and show $pG\cong pC_1\oplus \ldots \oplus pC_k.$}
\visible<6->{\item<beamer:check@12-> Show $pG\cong pH$}
\visible<7->{\item<beamer:check@13-> Apply the induction hypothesis to $pG$ and $pH$ and use it to rebuild $G$ and $H.$}
\ee



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