# The Unapologetic Mathematician

## Group actions

Okay, now we’ve got all the setup for one big use of group theory.

Most mathematical structures come with some notion of symmetries. We can rotate a regular $n$-sided polygon $\frac{1}{n}$ of a turn, or we can flip it over. We can rearrange the elements of a set. We can apply an automorphism of a group. The common thread in all these is a collection of reversible transformations. Performing one transformation and then the other is certainly also a transformation, so we can use this as a composition. The symmetries of a given structure form a group!

What if we paint one side of the above $n$-gon black and the other white, so flipping it over definitely changes it. Then we can only rotate. The rotations are the collection of symmetries that preserve the extra structure of which side is which, and they form a subgroup of the group of all symmetries of the $n$-gon. The symmetries of a structure preserving some extra structure form a subgroup!

As far as we’re concerned right now, mathematical structures are all built on sets. So the fundamental notion of symmetry is rearranging the elements of a set. Given a set $S$, the set of all bijections from $S$ to itself ${\rm Bij}(S)$ is a group. We’ve actually seen a lot of these before. If $S$ is a finite set with $n$ elements, ${\rm Bij}(S)$ is the symmetric group on $n$ letters.

Now, a group action on a set is simply this: a homomorphism from $G$ to ${\rm Bij}(S)$. That is, for each element $g$ in $G$ there is a bijection $p_g$ of $S$ and $p_{gh}(x)=p_g(p_h(x))$.

It’s important to note what seems like a switch here. It’s really due to the notation, but can easily seem confusing. What’s written on the right is “do the permutation corresponding to $h$, then the one corresponding to $g$“. So we have to think of the multiplication in $G$ as “first do $h$, then do $g$“.

In what follows I’ll often write $p_g(x)$ as $gx$. The homomorphism property then reads $(gh)x=g(hx)$

I’ll throw out a few definitions now, and follow up with examples in later posts.

We can slice up $S$ into subsets so that if $x$ and $x'$ are in the same subset, $x'=gx$ for some $g$, and not if they’re not in the same subset. In fact, this is rather like how we sliced up $G$ itself into cosets of $H$. We call these slices the “orbits” of the $G$ action.

As an important special case of the principle that fixing additional structure induces subgroups, consider the “extra structure” of one identified point. We’re given an element $x$ of $S$, and want to consider those transformations in $G$ which send $x$ to itself. Verify that this forms a subgroup of $G$. We call it the “isotropy group” or the “stabilizer” of $x$, and write it $G_x$.

I’ll leave you to ponder this: if $x$ and $x'$ are in the same $G$-orbit, show that $G_x$ and $G_{x'}$ are isomorphic.

February 20, 2007 -

1. You really want to repair that missing closing sub-tag in the end — it changes the layout of EVERYTHING else on the first page.

Comment by Mikael Johansson | February 21, 2007 | Reply

2. Very odd.. I accidentally typed “sup” instead of “sub”, but Safari evidently knew what I meant…

Comment by John Armstrong | February 21, 2007 | Reply

3. [...] “Tale of Groupoidification”. He really fleshes out this extension of the notion of group actions, and ties it into the concept of spans. I hope it’s not gushing too much to say that spans [...]

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4. [...] The analogue in ring theory for the idea of a group action is that of a module. Again we want every element of the ring to behave like a function on a set and [...]

Pingback by Modules « The Unapologetic Mathematician | April 21, 2007 | Reply

5. [...] Finds. He continues his “Tale of Groupoidification”. It features a great comparison of group actions on sets and those on vector spaces (which we’ll get to soon enough). Even better, he gives a [...]

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6. [...] and group actions — categorically The theory of group actions looks really nice when we translate it into the language of categories. That’s what I plan to [...]

Pingback by Groups and group actions — categorically « The Unapologetic Mathematician | June 8, 2007 | Reply

7. [...] is still beyond us at this point, but there are others. One nice place groupoids come up is from group actions. In fact, Baez is of the opinion that the groupoid viewpoint is actually more natural than the [...]

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8. [...] point we’ll call so that . It’s straightforward from here to show that this gives an action of the vector space (considered as an abelian group) on the affine space [...]

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9. [...] a set with some binary operation defined on it, sure, but what does it do? We’ve seen groups acting on sets before, where we interpret a group element as a permutation of an actual collection of elements. [...]

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10. [...] here’s the upshot: the general linear group acts on the hom-set by conjugation — basis changes. In fact, this is a representation of the group, [...]

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11. [...] group on the vector space of matrices over by conjugation. What we want to consider now are the orbits of this group action. That is, given two matrices and , we will consider them equivalent if there [...]

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12. [...] Ising model of ferromagnetism has found some evidence they claim is linked to certain symmetry; an action of the Lie group on the space of states which preserves some interesting property or another. I [...]

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13. [...] Weyl Group on Weyl Chambers With our latest lemmas in hand, we’re ready to describe the action of the Weyl group of a root system on the set of its Weyl chambers. Specifically, the action is [...]

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14. [...] to existence. Given a vector , we consider its orbit — the collection of all the as runs over all elements of . We have to find a vector in this [...]

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15. [...] Ising model of ferromagnetism has found some evidence they claim is linked to certain symmetry; an action of the Lie group on the space of states which preserves some interesting property or another. I [...]

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16. [...] detailed subject. And since every finite group can be embedded in a permutation group — its action on itself by left multiplication permutes its own elements — and many natural symmetries come [...]

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17. [...] and Representations From the module perspective, we’re led back to the concept of a group action. This is like a -module, but “discrete”. Let’s just write down the axioms for [...]

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18. [...] now multiplication on the left by shuffles around the cosets. That is, we have a group action of on the quotient set , and this gives us a permutation representation of [...]

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