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text: 'The notion that group theory captures the idea of "symmetry" derives from the notion of the symmetric group, and the very important theorem due to Cayley that every group is a subgroup of a symmetric group.\n\n# Definition\n\nLet $X$ be a [-3jz]. A [499 bijection] $f: X \\to X$ is a *permutation* of $X$.\nWrite $\\mathrm{Sym}(X)$ for the set of permutations of the set $X$ (so its elements are functions).\n\nThen $\\mathrm{Sym}(X)$ is a group under the operation of composition of functions; it is the *symmetric group on $X$*.\n(It is also written $\\mathrm{Aut}(X)$, for the *automorphism group*.)\n\nWe write $S_n$ for $\\mathrm{Sym}(\\{ 1,2, \\dots, n\\})$, the *symmetric group on $n$ elements*.\n\n# Elements of $S_n$\n\nWe can represent a permutation of $\\{1,2,\\dots, n\\}$ in two different ways, each of which is useful in different situations.\n\n## Double-row notation\n\nLet $\\sigma \\in S_n$, so $\\sigma$ is a function $\\{1,2,\\dots,n\\} \\to \\{1,2,\\dots,n\\}$.\nThen we write $$\\begin{pmatrix}1 & 2 & \\dots & n \\\\ \\sigma(1) & \\sigma(2) & \\dots & \\sigma(n) \\\\ \\end{pmatrix}$$\nfor $\\sigma$.\nThis has the advantage that it is immediately clear where every element goes, but the disadvantage that it is quite hard to see the properties of an element when it is written in double-row notation (for example, "$\\sigma$ cycles round five elements" is hard to spot at a glance), and it is not very compact.\n\n## Cycle notation\n\n[49f Cycle notation] is a different notation, which has the advantage that it is easy to determine an element's order and to get a general sense of what the element does.\nEvery element of $S_n$ [49k can be expressed in (disjoint) cycle notation in an essentially unique way].\n\n## Product of transpositions\n\nIt is a useful fact that every permutation in a (finite) symmetric group [4cp may be expressed] as a product of [4cn transpositions].\n\n# Examples\n\n- The group $S_1$ is the group of permutations of a one-point set. It contains the identity only, so $S_1$ is the trivial group.\n- The group $S_2$ is isomorphic to the [-47y] of order $2$. It contains the identity map and the map which interchanges $1$ and $2$.\n\nThose are the only two [3h2 abelian] symmetric groups.\nIndeed, in cycle notation, $(123)$ and $(12)$ do not commute in $S_n$ for $n \\geq 3$, because $(123)(12) = (13)$ while $(12)(123) = (23)$.\n\n- The group $S_3$ contains the following six elements: the identity, $(12), (23), (13), (123), (132)$. It is isomorphic to the [-4cy] $D_6$ on three vertices. ([group_s3_isomorphic_to_d6 Proof.])\n\n# Why we care about the symmetric groups\n\nA very important (and rather basic) result is [49b Cayley's Theorem], which states the link between group theory and symmetry.\n\n%%%knows-requisite([4bj]):\n# Conjugacy classes of $S_n$\n\nIt is a useful fact that the conjugacy class of an element in $S_n$ is precisely the set of elements which share its [4cg cycle type]. ([4bh Proof.])\nWe can therefore [4bk list the conjugacy classes] of $S_5$ and their sizes.\n%%%\n\n# Relationship to the [-4hf]\n\nThe [-4hf] $A_n$ is defined as the collection of elements of $S_n$ which can be made by an even number of [4cn transpositions]. This does form a group ([4hg proof]).\n\n%%%knows-requisite([4h6]):\nIn fact $A_n$ is a [-4h6] of $S_n$, obtained by taking the quotient by the [4hk sign homomorphism].\n%%%',
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