{"id":562,"date":"2007-12-13T17:28:09","date_gmt":"2007-12-13T17:28:09","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2007\/12\/13\/symmetric-groups-and-group-actions\/"},"modified":"2007-12-13T17:28:09","modified_gmt":"2007-12-13T17:28:09","slug":"symmetric-groups-and-group-actions","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2007\/12\/13\/symmetric-groups-and-group-actions\/","title":{"rendered":"Symmetric Groups and Group Actions"},"content":{"rendered":"

In my last post on group theory, I screwed up a bit in presenting an example. The example was using a pentagram as an illustration of something called a permutation group<\/em>. Of course, in
\nmy attempt to simplify it so that I wouldn’t need to spend a lot of time explaining it, I messed up. Today I’ll remedy that, by explaining what permutation groups – and their more important cousins, the symmetry groups<\/em> are, and then using that to describe what a group action is, and how the group-theory definition of symmetry can be applied to things that aren’t groups.<\/p>\n

<\/p>\n

As I alluded to in the last post, permutation groups are very fundamental. You’ll see part of why
\nthat is later in this post. But there’s also a historical reason. Group theory was developed as
\na part of the algebraic study of equations. One of the main occupations of people studied algebra
\nup to the 19th century was finding equations to compute the roots of polynomials. So, for
\nexample, anyone who’s taken any high school math knows the quadratic equation, which can be used
\nto find the roots of a quadratic polynomial.<\/p>\n

The quadratic solution has been known for a very long time. There are records dating back to the
\nBabylonians that contain forms of the quadratic equation. It took a ridiculously long time to get from there to a general solution for cubics. Quartics followed very soon after cubics. But then, after the quartic solution, there was a couple of hundred years of delay with no progress. Neils Henrik Abel and Evariste Galois, both very young and very unlucky mathematicians, roughly simultaneously proved
\nthat there was no general solution for polynomials of degree five. Galois did it by working out
\nsymmetry properties of the solutions of polynomials – which come from the permutation groups
\nof those solutions – and showing that there was no possible way to get a solution because of the properties of those groups. We’ll leave it at that for now; later, I’ll come back to that, and show how you can form permutation groups from the solutions of a polynomial, and how the structure of the permutation groups can show that there are no algebraic solutions for orders greater than 4.<\/p>\n

Getting back on topic: what is a permutation group? Given a set of objects, O, a permutation
\nis a one-to-one mapping from O to itself. It defines a way of re-arranging the elements of the set. So, for example, given the set of numbers {1, 2, 3}, a permutation of them is {1→2, 2→3, 3→1}. A permutation group<\/em> is a set of permutations over a set, with the composition
\nof permutations as the group operator. So, for example, working with the set {1,2,3} again, the elements of the largest permutation group are:<\/p>\n

{ { 1→1, 2→2, 3→3 }, { 1→1, 2→3, 3→2 }, { 1→2, 2→1, 3→3 }, { 1→2, 2→3, 3→1 }, { 1→3, 2→1, 3→2 }, { 1→3, 2→2, 3→1 } }<\/p>\n

To see the group operation, let’s take two values from the set. Let f={1→2, 2→3, 3→1}, and let g={1→3, 2→2, 3→1}. Then the group operation of function composition will generate the result: fˆg={1→2, 2→1, 3→3}.<\/p>\n

The identity of the group is obvious: 1O<\/sub> = {1→1, 2→2, 3→3}. Inverses are also obvious: just reverse the direction of the arrows: { 1→3, 2→1, 3→2 }-1<\/sup> =
\n{ 3→1, 1→2, 2→3 }.<\/p>\n

When you take the set of all<\/em> permutations over a collection of N values, the result is the
\nlargest possible permutation group over those values. That group is called the symmetric group<\/em>
\nof size N, or S<\/b>N<\/sub>. The symmetric group is fundamental: every finite group is a subgroup of a finite symmetric
\ngroup; which in turn means that every possible symmetry of every possible group is embedded in the structure of the corresponding symmetric group. <\/p>\n

To formalize that just a tad, I’ll need to formally define a subgroup. Fortunately, that’s quite
\neasy. If you have a group (G,+), then a subgroup of it is a group (H,+) where H⊆G. In english,
\na subgroup is a subset of the values of a group, using the same<\/em> group operator, and which
\nsatisfies the required properties of a group. So, for instance, the subgroup needs to be closed under
\nthe group operator.<\/p>\n

For example, if you have the group of integers, with addition of its operation, then the set of even<\/em> integers in a subgroup. Any time you add any two even integers, the result is an even integer. Any time you take the inverse of an even integer, it’s still even. So it’s closed. You can work through the other properties, and it will satisfy all of them.<\/p>\n

There’s a stronger form of subgroup, called a normal<\/em> subgroup. A normal subgroup (H,+) of a
\ngroup (G,+) is a subgroup that satisfies one additional property: ∀x∈G: ∀y:∈D
\nx+y+x-1<\/sup>∈H. That looks<\/em> like something that should be obviously true for all
\nsubgroups. It isn’t. The reason that it looks obvious is that we intuitively expect<\/em> the group
\noperator to be commutative. But our definition of groups does not<\/em> require the group operator to
\nbe commutative. There are many groups whose group operators are<\/em> commutative: they’re called the
\nAbelian<\/em> groups. But there are also many that aren’t. All subgroups of an abelian group
\nare<\/em> normal. But there are subgroups of non-abelian groups that are not normal.<\/p>\n

We’re almost done with the definitions. But there’s a couple easy ones that I need
\nbefore I can explain group operators.<\/p>\n

A trivial<\/em> group is a group which contains only<\/em> an identity value. A simple group is basically sort-of the group-wise equivalent of a prime number: a simple<\/em> group is a group whose only normal subgroups are the trivial group, and the group itself.<\/p>\n

Ok, now we’re finally ready. As I’ve talked about before, a group defines a kind of symmetry, otherwise known as an immunity to some kind of transformation. But we don’t want to have to define groups and group operators for every set of values that we see as symmetric. What we’d like to do is capture the fundamental idea of a kind of symmetry using the simplest group that really
\nexhibits that kind of symmetry, and the able to use<\/em> that group as the definition of
\nthat kind of symmetry. To do that, we need to be able to describe what it means to apply the
\nsymmetry defined by a group to some set of values. We call the transformation of a set produced
\nby applying a symmetric transformation defined by a group G as the group action<\/em> of the group G.<\/p>\n

Suppose we want to apply a group G as a symmetric transformation on a set A. What we can
\ndo is take the set A, and define the symmetric group over A, S<\/b>A<\/sub>. Then we can
\ndefine a mapping – to be more precise, a homomorphism – from the group G to SA<\/sub>. That
\nhomomorphism is the action<\/em> of G on the set A. To make that formal:<\/p>\n

If (G,+) is a group, and A is a set, then the group action<\/em> of G on A is a function f such that:<\/p>\n

    \n
  1. ∀g,h∈G: (∀a∈A : f(g+h,a) = f(g,f(h,a)))<\/li>\n
  2. ∀a∈A: f(1G<\/sub>,a) = a.<\/li>\n<\/ol>\n

    All of which says that if you’ve got a group defining a symmetry, and a set you want to apply a symmetric transformation to, then there’s a way o mapping from the elements of the group to the elements of the set, and you can perform the symmetric group operation through<\/em> that map. The group operation is an application of the group operation through that mapping.<\/p>\n

    Every symmetric operation can be characterized by some kind of group; and using that group’s group operation, that symmetric operation can be applied to any desired set of values. So we can, for example, use the group of addition over the real numbers to define mirror symmetry on a two dimensional image.<\/p>\n","protected":false},"excerpt":{"rendered":"

    In my last post on group theory, I screwed up a bit in presenting an example. The example was using a pentagram as an illustration of something called a permutation group. Of course, in my attempt to simplify it so that I wouldn’t need to spend a lot of time explaining it, I messed up. […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[26],"tags":[],"class_list":["post-562","post","type-post","status-publish","format-standard","hentry","category-group-theory"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-94","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/562","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=562"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/562\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=562"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=562"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=562"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}