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Is abelian group <span class='MathJax_Preview'><img src='http://moebiuscurve.com/wp-content/plugins/latex/cache/tex_dfcf28d0734569a6a693bc8194de62bf.gif' style='vertical-align: middle; border: none; ' class='tex' alt="G" /></span><script type='math/tex'>G</script> always isomorphic to H×(G/H)?
Let $$G= \mathbb{Z}_4=\mathbb{Z}/4\mathbb{Z}$$ and $$H=\langle2\rangle$$ Then $$G= \mathbb{Z}_4 = \mathbb{Z}/4 \mathbb{Z} = \{0,1,2,3\}$$.We know $$\mathbb{Z}/4\mathbb{Z}$$ is cyclic. Cayley Table of $$G=\mathbb{Z}/4 \mathbb{Z}$$ is:$$\begin{array}{|c|c|c|c|c|}\hline\hline\textbf{+} & \textbf{0} & \textbf{1} & \textbf{2} & \textbf{3}\\\hline \textbf{0} & 0 & 1 & 2 & 3 \\\hline \textbf{1} & 1 & 2 & 3 & 0 \\\hline \textbf{2} & 2& 3 & 0 & 1\\\hline\textbf{3} & 3 & 0 & 1 & 2 \\\hline \end{array}$$ $$\cong$$ $$\begin{array}{|c|c|c|c|c|}\hline\hline\bf{+}…