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I'm new to Abstract Algebra and I'm trying to explain myself what a group really is. So far I've looked into a couple of books, read a little about groups here and there and this is what I've understood so far.

Symmetry: A symmetry of an object $O$ is a bijective function $s:O \to O$ i.e. $s(O)=O$

Intuitively, they are transformations that leave the shape of the object unchanged, they just shuffle around the pieces of the object.

Here, I've taken the object $O$ to be the set $O$ since it is my understanding that everything is a set.

Group: A group $G$ is collection of symmetries of the object $O$ satisfying:

  1. It has a symmetry which does nothing to the object $O$
  2. Every symmetry applied to $O$ can be reversed.
  3. Any sequence of symmetries applied to $O$ is also a symmetry of $O$ present in $G$
  4. Given a sequence of symmetries i.e. $s_1,s_2, s_3$, the order in which you apply this sequence of symmetries to $O$ doesn't matter i.e. $(s_1 \cdot s_2) \cdot s_3 = s_1 \cdot (s_2 \cdot s_3)$

Hence a Group is merely a "map" or some kind of "guide book" that tells you "how" the symmetries interact with each other and you can just throw away the object, you don't care about it.

Is my intuition so far correct? If there are any flaws in it, please point out. If you can improve on this intuition, please by all means.

Now my questions are,

  1. Why do you not care about the object?
  2. Given a group, how can I find the object that this group describes the symmetries of?
  3. Does every group describe symmetries of some object?
glS
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William
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  • For each point there are several interesting posts at this site. Concerning 3. for example this post, and in general for example this one. I think it makes sense to include these posts into the discussion (also the MO post linked in the answer!). So far, you've only "looked into a couple of books", as you said. Take a look at the discussions at this site, too. – Dietrich Burde Jan 28 '21 at 11:01
  • @DietrichBurde I've only "looked into a couple of books" and read here and there but yea I searched "group intuition" here and it shows some weird results about lie groups and some other advance stuff and nothing on groups' intuition at a beginner level hence I decided to ask this. – William Jan 28 '21 at 11:22
  • But there are several elementary posts here, right? I have found many other posts about your questions. Have you seen this one, for example? Or this one? I think there are already really good answers here. – Dietrich Burde Jan 28 '21 at 11:26

2 Answers2

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Your intuition is good, but there are some problems. I must admit that I have no idea about the meaning of “Any sequence of symmetries applied to $O$ is also a symmetry of $O$ present in $G$”. Perhaps that you meant that the composition of symmetries present in $G$ also belongs to $G$.

Note that every group $G$ is a set of symmetries of $G$ itself: for every $h\in G$, consider the map$$\begin{array}{ccc}G&\longrightarrow&G\\g&\mapsto&hg.\end{array}$$The set of all these maps is a set of symmetries of $G$. So, yes, in this sense every group is the group of symmetries of some object.

However, thinking this way will perhaps make harder to see that quite often the same group is a group of symmetries of two very distinct objects. For instance, the group of all isometries of a regular triangle and the group of all permutations of the set $\{1,2,3\}$ are the same group (technically speaking, they are isomorphic), although, of course, we have very distinct objects here.

José Carlos Santos
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  • "Your intuition is good" but I not great, I guess? Can you improve on it? By "Any sequence of symmetries applied to $O$ is also a symmetry of $O$ present in $G$", I meant given a sequence of symmetries $s_1,s_2, \ldots, s_n$ applying $s_1, s_2, \ldots, s_n$ to $O$ leaves the shape of $O$ unchanged, I was trying to capture the closure property as you already know. I worded it that way because I was trying to avoid big words like "composition" and use laymen language. – William Jan 28 '21 at 10:53
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    Of course that the shape of $O$ remains unchanged; otherwise, at least one of them would not be symmetry of $O$, right?! What matters here is that, after you apply all those symmetries, you still have an element of $G$. – José Carlos Santos Jan 28 '21 at 10:56
  • "Quite often the same group is a group of symmetries of two very distinct objects." That is a very good point. I just thought about it and I came up with the set of counterclockwise rotations ${r_{0}, r_{90}, r_{180}, r_{270} }$ is group of symmetries of both, a square and a circle. Lovely! – William Jan 28 '21 at 10:57
  • Yes and that's why I said, "... present in $G$" in my original question. – William Jan 28 '21 at 10:59
  • Oh and sorry, I completely forgot to ask, do you have a better way to imagine groups/better intuition than this? – William Jan 28 '21 at 17:02
  • I think that it is more useful to see a group as a set of bijections from a set into itself, but not necessarily as a set of symmetries of a geometric object. – José Carlos Santos Jan 28 '21 at 17:54
  • Sorry to bother again, but the mapping you give that is for any $h \in G, h:G \to G$ given by $h(g)=hg$ How do you show that it is bijective? (Since it's a symmetry, it's bijective). Now $hg_1=hg_2 \Rightarrow g_1=g_2$ but how do you show it is surjective? That is for each $ g \in G, \exists g_0$ such that $g=hg_0$? – William Jan 29 '21 at 20:39
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    Simply take $g_0=h^{-1}g$. – José Carlos Santos Jan 29 '21 at 23:53
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Is my intuition so far correct?

Pretty much, yeah!

Why do you not care about the object? Two reasons:

  1. Because groups are, by themselves, interesting objects.
  2. Because by not caring about the particular object, everything you discover about its group of symmetries holds for every object with the same symmetry group!

Given a group, how can I find the object that this group describes the symmetries of? Does every group describe symmetries of some object?

It's... Complicated.

5xum
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  • "everything you discover about its group of symmetries holds for every object with the same symmetry group!" Interesting. Is this a clever way to say that, that group of symmetries is isomorphic to the object's symmetry group $S_n$? But 1) What happens when the group is infinite?$S_n$ makes sense for only for a finite group. 2) How do you find out the symmetry group of an object? – William Jan 28 '21 at 12:35