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Symmetric group:S5

39 bytes added, 00:45, 19 August 2012
Arithmetic functions
| {{arithmetic function value order|120}} || As <math>\! S_n, n = 5:</math> <math>\! n! = 5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120</math> <br><br> As <math>\! PGL(2,q), q = 5</math> (see [[order formulas for linear groups of degree two]]): <math>\! q^3 - q = q(q-1)(q+1) = 5^3 - 5 = 5(4)(6) = 120</math><br>As <math>\! P\Gamma L(2,q), q = p^r, q = 4, p = 2, r = 2</math> (see [[order formulas for linear groups of degree two]]): <math>\! r(q^3 - q) = 2(4^3 - 4) = 2(60) = 120</math>.
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| {{arithmetic function value exponent given order|60|120}} || As <math>\! S_kS_n, k n = 5:</math> <math>\! \operatorname{lcm} \{ 1,2,\dots,n \} = \operatorname{lcm} \{1,2,3,4,5 \} = 60</math> <br><br> As <math>\! PGL(2,q), q = p = 5:</math> (where <math>\! p</math> is the underlying prime for <math>\! q</math>) <math>\! p(q^2 - 1)/2 = 5 \cdot (5^2 - 1)/2 = 60</math>
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| [[derived length]] || -- || || not a solvable group.
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