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What kind of “symmetry” is the symmetric group about?

Could you tell me please, why Symmetric group is called "symmetric"?

I found an example with quadrate, where it's explained that if we rotate it it's still symmetric. But I still don't get, why group of permutations is called symmetric.

Best.

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marked as duplicate by lhf, Jonas Meyer, t.b., Jack Schmidt, Zev Chonoles Dec 27 '11 at 10:54

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3 Answers 3

This terminology can, at least partially, be traced back to Lagrange and then Cauchy and Galois, who looked for solutions of polynomial equations of higher degree. Note that every symmetric group acts on the variables of a multi-variate function through $$ f^\sigma:(x_1,\ldots,x_n)\mapsto f(x_{\sigma(1)},\ldots,x_{\sigma(n)}). $$ The functions $f$ left invariant by every $\sigma$ in the corresponding symmetric group are exactly the so-called symmetric functions.

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A point of view proposed by Klein in his famous Erlangen program was that geometry should be studied via the group of symmetries of the space in question, that is, the set of transformations that preserve whatever structure the space has.

In the absence of any structure, the symmetries of a set $X$ are just the bijections $X\to X$.

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I don't know if there's an actual reason, but I know this : one of the basic reasons for the existence of groups in many fields of science is to study symmetries of objects. There is this theorem that justifies the use the term "symmetric" for $S_A$ :

Cayley's theorem. Let $(G,\cdot)$ be a group. Then $(G,\cdot)$ is isomorphic to a subgroup of $S_G$, where $S_G$ is the group of all bijections from $G$ to $G$ (here $G$ is considered the underlying set and $(G,\cdot)$ is the group, for clarification).

So in some way any group is a subgroup of $S_A$ for some set $A$, hence any symmetry over some object represented by a group structure lies "somewhere in $S_A$ for some $A$. That is why I agree with the name "symmetric group" and I think it should stay that way, but I have no idea why historically we have been naming it like that.

Hope that helps,

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