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text: '[summary:\nExamples of [-3gd groups], including the [-497 symmetric groups] and [-general_linear_group general linear groups].\n]\n\n# The symmetric groups\n\nFor every positive integer $n$ there is a group $S_n$, the [497 symmetric group] of order $n$, defined as the group of all permutations (bijections) $\\{ 1, 2, \\dots n \\} \\to \\{ 1, 2, \\dots n \\}$ (or any other [-3jz] with $n$ elements). The symmetric groups play a central role in group theory: for example, a [3t9 group action] of a group $G$ on a set $X$ with $n$ elements is the same as a [47t homomorphism] $G \\to S_n$. \n\nUp to [4bj conjugacy], a permutation is determined by its [4cg cycle type]. \n\n# The dihedral groups\n\nThe [4cy dihedral groups] $D_{2n}$ are the collections of symmetries of an $n$-sided regular polygon. It has a [5j9 presentation] $\\langle r, f \\mid r^n, f^2, (rf)^2 \\rangle$, where $r$ represents rotation by $\\tau/n$ degrees, and $f$ represents reflection. \n\nFor $n > 2$, the dihedral groups are non-commutative.\n\n# The general linear groups\n\nFor every [481 field] $K$ and positive integer $n$ there is a group $GL_n(K)$, the [general_linear_group general linear group] of order $n$ over $K$. Concretely, this is the group of all invertible $n \\times n$ [matrix matrices] with entries in $K$; more abstractly, this is the [automorphism automorphism group] of a [3w0 vector space] of [vector_space_dimension dimension] $n$ over $K$. \n\nIf $K$ is [algebraically_closed_field algebraically closed], then up to conjugacy, a matrix is determined by its [Jordan_normal_form Jordan normal form]. ',
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