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What is the difference between a permutation group and a symmetric group?

I know such questions asked before on Math Stack Exchange, for example here.

But I would appreciate it if someone could spell out the difference without discussing group actions.

Shaun
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F. A. Mala
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    Each subgroup of a symmetric group is called a permutation group – Wuestenfux Jul 30 '22 at 08:19
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    We have ${\rm Sym}(X)={ f\colon X\rightarrow X\mid \text{f is a bijection}}$ for a set $X$. If $X={1,2,\ldots ,n}$, then it is called $S_n$, the symmetric group on $n$ letters. Wikipedia spells it out without discussing group actions. In the same way, look up permutation group in wikipedia. – Dietrich Burde Jul 30 '22 at 08:41
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    What is wrong with my question? Why would anyone vote it down? – F. A. Mala Jul 30 '22 at 08:46
  • Given a positive integer $n$, the set of all permutations of $n$ things forms a group, under composition, called the symmetric group $S_n$. So, the symmetric group is a group of permutations. Also, it follows that every subgroup of the symmetric group is (or can be viewed as) a group of permutations. Indeed, every finite group is isomorphic to a subgroup of $S_n$, so every finite group can be viewed as a group of permutations. – Gerry Myerson Jul 30 '22 at 10:04
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    The answer is very simple: a permutation group is a subgroup of a symmetric group. – Derek Holt Jul 30 '22 at 10:47

1 Answers1

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A symmetric group on a set $X$ is the set of all bijections on $X$ (called permutations) under the operation of function composition.

A permutation group is a subgroup of a symmetric group. Be careful, however, because Cayley's Theorem states that every group is isomorphic to a subgroup of a symmetric group; so to be clear: a permutation group is a group of permutations, not necessarily equal to a symmetric group.

For example, $G=(\{e, (13)\}, \circ)$ is a permutation group, since it is a subgroup of the symmetric group $S_3$. But $H=(\{1,-1\},\times)\cong G$, despite $H$ not being a permutation group.

Shaun
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