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Why are groups required to have inverses? What is the motive behind it? It doesn't fall out of another requirement, so what was the goal of adding them in?

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    $\begingroup$ People do study things like groups that do not have inverses, they are called monoids. One reason groups are nice is that equations equations of the form, $ax=b$ have unique solutions. $\endgroup$ – Baby Dragon Sep 25 '13 at 1:07
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    $\begingroup$ Groups "existed" before we named them. People realized there were an bunch of examples of what we now call "groups", that is sets with an associative operation, a unity and inverses, and decided to call them like that. It wasn't just pulled out of thin air! =D $\endgroup$ – Pedro Tamaroff Sep 25 '13 at 1:19
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    $\begingroup$ So for example, you have the group of injective maps of a set, you have the group of Euclidean motions, you have the group of roots of unity, you have the group of rotations and reflections of $n$-agons, and so forth. $\endgroup$ – Pedro Tamaroff Sep 25 '13 at 1:21
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1) You don't have to require the existence of inverses. If you have a set $M$ with a binary operation that is associative and contains an identity, you have what is called a monoid. Monoids are very interesting structures that come up in many areas of mathematics.

2) Why are groups more interesting than monoids? (At least, to most people's tastes.) One reason is that groups are related to the notion of symmetry. A symmetry of an object is a (bijective) transformation that leaves the object unchanged. In that case, the inverse of the transformation will also leave the object unchanged. So the algebraic structure that models symmetries is associative (composition of transformations is associative), has an identity (doing nothing is always a symmetry!), and has inverses. So groups are the right algebraic structure to study symmetry.

3) Mathematically, the existence of inverses is very powerful. If you try to solve the equation $ax = b$ in a monoid ($a,b$ are given; $x$ is unknown), then you may no solutions, one solution, or many solutions. But in a group, there is always a unique solution: $x = a^{-1} b$. This is very useful. Said differently, in a group if I perform some unknown operation $x$ followed by $a$ and the resulting composition is the operation $b$, then I can figure out what $x$ had to be. If instead we were working in a monoid, we would not be able to recover what $x$ is (or even know whether or there is some $x$ for which the statement holds), without additional information specific to the monoid at hand.

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  • $\begingroup$ I think groups are more interesting than monoids. What are some great theorems about monoids?? $\endgroup$ – lhf Sep 25 '13 at 1:28
  • $\begingroup$ Relevant post by Pete Clark $\endgroup$ – Zev Chonoles Sep 25 '13 at 1:33
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    $\begingroup$ Their are not really that many great theorems about naked groups. Their are some nice theorems about groups with properties, such as finite groups, nilpotent groups, or other adjective groups. $\endgroup$ – Baby Dragon Sep 25 '13 at 1:57
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Consider you live in a space where you can scale things. Usually, when something has scale 0 it's not considered the same shape at a different size, and all things that are just different sizes of things should be able to be scaled back up to any other scale, including the original. If the space you're talking about is a line, and there's exactly one point in the space at every scale from the origin, you're talking about a divisible operation on the line, except at the origin. Likewise, when higher-dimensional spaces have reversible scaling and have an underlying coordinate system, this will likely translate into a reversible multiplication on those coordinates.

Consider you live in a space where if you can move from point A to point B, then you can move from point B to point A. This is a divisible operation on the points, at least if associativity exists - otherwise you might still be able to move through one point to reach another, though you can't go directly from one to the other.

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