Before I go back to the adventures with tetrahedrane, I thought I would take a short break to talk about group theory.
I started learning group theory seriously some time back, as a part of chemistry course. Till now I had seen notations like \(D_{h}\)and \(T_{d}\) and had secretly wondered what they were with a vague feeling that they somehow represented what the molecule looked like. My footing in rigorous math is at best questionable, so pardon me wherever I might not be that precise.
Groups are an abstract collections of things/objects (as it might be evident from the name) with a particular operations defines between elements (like addition or multiplication) that obey particular properties. The most obvious and oldest of groups are things known as rotation groups. They are what they sound like: groups which are collection of all the rotations you can do. For example Euler's theorem says that any rotations of an object be it a chair or a planet where a point in the body remains fixed ( for example if you let a coin spin, then the points on its axes remain fixed) can be described by three rotations about a set of axes/ three Euler angles. More on rotations later. So lets get back to properties that defines a group.
Suppose \(G=\left\{ g_{1},g_{2},...g_{n} \right\} \) is our would be group with the operation *. For it to be a group it should satisfy the following properties.
1. Identity - There should be an element \(e\) in \(G\) such that for any element \(g\) in \(G\), \( g*e=e*g=e\)
2.Inverse - For every element \(g\) in \(G\), there should be another element \(a\) in \(G\) such that \(a*g=g*a=e\)
3. Closure - For any two elements \(a\) and \(b\) in \(G\), their product should always be in G itself.
4.Associativity - \(a*(b*c)=(a*b)*c\) for any \(a,b,c\) in \(G\).
I started learning group theory seriously some time back, as a part of chemistry course. Till now I had seen notations like \(D_{h}\)and \(T_{d}\) and had secretly wondered what they were with a vague feeling that they somehow represented what the molecule looked like. My footing in rigorous math is at best questionable, so pardon me wherever I might not be that precise.
Groups are an abstract collections of things/objects (as it might be evident from the name) with a particular operations defines between elements (like addition or multiplication) that obey particular properties. The most obvious and oldest of groups are things known as rotation groups. They are what they sound like: groups which are collection of all the rotations you can do. For example Euler's theorem says that any rotations of an object be it a chair or a planet where a point in the body remains fixed ( for example if you let a coin spin, then the points on its axes remain fixed) can be described by three rotations about a set of axes/ three Euler angles. More on rotations later. So lets get back to properties that defines a group.
Suppose \(G=\left\{ g_{1},g_{2},...g_{n} \right\} \) is our would be group with the operation *. For it to be a group it should satisfy the following properties.
1. Identity - There should be an element \(e\) in \(G\) such that for any element \(g\) in \(G\), \( g*e=e*g=e\)
2.Inverse - For every element \(g\) in \(G\), there should be another element \(a\) in \(G\) such that \(a*g=g*a=e\)
3. Closure - For any two elements \(a\) and \(b\) in \(G\), their product should always be in G itself.
4.Associativity - \(a*(b*c)=(a*b)*c\) for any \(a,b,c\) in \(G\).
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