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  "body": "\n## Example\n\nLet $F$ be a field and let $n \\ge 2$ be an integer.\nThe *projective special linear group* is the [[quotient group|0KYX2N69J0]]\n$$PSL_n(F) = SL_n(F) / Z,$$\nwhere $SL_n(F)$ is the [[special linear group|0V9PMN0RBH]] and $Z$ is its [[centre|0DZ93493VN]].\nThe centre $Z$ consists of the scalar matrices $\\lambda I_n$ with $\\lambda^n = 1$.\nFor a [[finite field|0ZXZ82FTX2]] $\\F_q$ with $q$ elements, write $PSL_n(q)$.\n\nTake $n = 2$ and $q = 7$.\nThe order of $SL_2(7)$ is $q(q^2 - 1) = 7 \\cdot 48 = 336$.\nThe scalars $\\lambda I_2$ with $\\lambda^2 = 1$ are $\\pm I_2$, so $\\abs{Z} = 2$.\nHence $PSL_2(7)$ has order $168$.\nIt is a [[simple group|0YMYMNQ0QZ]], and it is the second smallest non-abelian simple group, after the alternating group $A_5$ of order $60$.\n\n## Remarks\n\nThe group $PSL_n(q)$ has order\n$$\\frac{1}{\\gcd(n, q - 1)} \\, q^{n(n-1)/2} \\prod_{i=2}^{n} (q^i - 1).$$\nIt is simple for all $n \\ge 2$ and all prime powers $q$, except $PSL_2(2) \\cong S_3$ and $PSL_2(3) \\cong A_4$.\nThese groups form one of the infinite families of [[groups of Lie type|NPEBABQ49M]] in the [[classification of finite simple groups|06J6P03MG7]].\nThe group $PSL_n(F)$ acts faithfully on the projective space of lines in $F^n$, which explains the name.\n\n## Explanations\n\n### L1 human\n\nStart with the $n \\times n$ matrices of determinant $1$ over a field.\nThen declare two matrices equal when one is a scalar multiple of the other.\nWhat is left is the projective special linear group.\nOver finite fields these groups give an endless supply of simple groups, the building blocks of all finite groups.\nThe smallest interesting one, $PSL_2(7)$, has $168$ elements and is the symmetry group of the Fano plane.\n",
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