In undergraduate abstract algebra we are not exposed to semi-direct products, so I was hoping someone could help me as I am doing some research in this area.(adsbygoogle = window.adsbygoogle || []).push({});

I am familiar with the definitions of direct products and normal groups, and I know that a semidirect product is one where one of the groups of the product is not normal. But I don't have a good intuitive understanding of what his means. To put it in more blunt terms, why should I care whether a group is normal or not?

I am working with the group of isometries in the taxicab metric, which is a semidirect product of T(2) and D_{4}. T(2) is the normal group in this case.

But let's start with something simpler. The Euclidean group of isometries of the plane is a semidirect product of O(2) and T(2). So all reflections, rotations, and transformations can be represented by combining the elements of these two groups. What does T(2) being "normal" to this group mean, in the geometric sense?

I hope my question makes sense. If not I will try and clarify.

-Dave K

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# Geometric understanding of Semi-direct product

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