Tag: Algebra

Is abelian group \(G\) 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{+}…

Some notes on Polynomials

(This is a work in progess…) (Note: I am doing some exploration on Galois Theory and I have written these notes on polynomials here for my own convenience. Please write in the comments section if any clarification is required. Also, I will not provide proofs here for the time being.) Some notations: The symbol \(\mathbb{C}\) is the domain of complex numbers; i.e. set of all possible complex numbers.The symbol \(\mathbb{R}\) is…

A Day out with FreePlane Mindmap and Mathematical Groups

Background I wanted to break away from Blender Learning/Modeling (Current passion) for a day or two. And so I began my exploration in Mathematical Groups using FreePlane Mind mapping software. I had recently switched from Freemind to to Freeplane. There was a simple enough reason for this decision – LaTeX support. Freeplane allows us to enter Mathematical equations using LaTeX. But a certain disadvantage of using Freeplane is that you…